Sparse Cyclic covering sets A283297 and A102508

[Ed Pegg]

I recently verified the first 31 terms of https://oeis.org/A283297  

These have the following number of solutions.  
{1, 1, 2, 1, 1, 4, 7, 4, 5, 2, 2, 25, 38, 16, 24, 9, 9, 186, 1, 150, 187, 74, 90, 36, 27, 1, 2088, 1095, 5}

The unique partition for 28 is {1, 3, 11, 5, 2, 6}, which is new to me.

However, for the value 95, listed with value 12, there is a conflict with
https://oeis.org/A102508  which has the value 11.

I have 10 partitions for 95 into 11 parts.
{{1,2,4,9,3,22,5,21,11,10,7}, {1,2,6,11,13,14,8,15,16,5,4}, {1,3,9,11,6,8,2,5,28,4,18},{1,3,9,11,20,13,6,8,2,5,17}, {1,3,12,7,4,18,2,10,25,8,5}, {1,3,12,13,2,8,14,19,7,11,5},{1,4,7,13,8,2,6,28,9,3,14}, {1,4,9,17,21,3,6,2,10,7,15}, {1,5,2,3,8,4,9,16,14,6,27},{1,8,3,22,18,6,2,5,10,4,16}}

[Ed Pegg]

I made an update to  A283297 . I'm mildly skeptical of some of the other over-70 values in A283297, but values are hard to verify. I'll start a search similar to my sparse ruler search.  At 
https://community.wolfram.com/groups/-/m/t/2663076  you can find my data sets.  
For Sparse Rulers, there's the pattern NJAS called "Dark Satanic Mills on a Cloudy Day".  
https://en.wikipedia.org/wiki/Sparse_ruler   
What's the pattern for A283297?  The natural break points are at a^2 + a + 1, so the 

splits will have lengths 1, 2, 4, 6, 8, 10, 12, 14, ... 

{{1},
 {2, 2},
 {3, 3, 3, 3},
 {4, 4, 4, 4, 4, 4},
 {5, 5, 5, 5, 5, 5, 6, 5},
 {6, 6, 6, 6, 6, 6, 6, 7, 7, 6},
 {7, 7, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8},
 {8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 8},
 {9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 9},
 {10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 10},
 {11, 12, 12, 11, 12, 12, 12, 12, 12}}

We can subtract n from the nth group

{{0},

{0,0},

{0,0,0,0},

{0,0,0,0,0,0},

{0,0,0,0,0,0,1,0},

{0,0,0,0,0,0,0,1,1,0},

{0,0,0,0,0,0,1,0,1,1,1,1},

{0,0,0,0,0,0,0,0,1,1,1,1,1,0},

{0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0},

{0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,0},

{0,1,1,0,1,1,1,1,1}}   (this line will not end in 0)

For the linear case, sparse rulers, the Brian Wichmann rulers and related cases made explorations not-too-horrible. A sparse partition for n=300 can apply to many other n values with simple modifications. But there are sporadic cases that are combinatorially challenging to find. For the Dark Mills pattern, it's unsolved whether the "Clouds" actually exist.

For the circular case, Singer's perfect partitions make almost everything sporadic. A perfect partition for n=211 seems to only be good for 211 specifically.  Still, there's lots of data and I have partition tweaking code that sometimes works. 

For n=94, partition {18, 1, 3, 9, 11, 6, 8, 2, 5, 28, 3} only misses 47. I'm not completely convinced an 11-partition for 94 is impossible, but I haven't found one.


 

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[Christian Sievers]

It took me a while to understand what your partitions have to do with the difference bases of  A283297.

I think in the examples section for that sequence, you should directly give the difference bases, or explain more.

[Ed Pegg]

I'm not exactly sure what a difference basis is.  I calculated a different sequence, and  A283297 had the identical values for the first 32 terms.

Length of partitions for n that cyclically cover all sums up to n. 
{{1}}
{{1, 1}}
{{1, 1, 2}}
{{1, 1, 3}, {1, 2, 2}}
{{1, 2, 3}}
{{1, 2, 4}}
{{1, 1, 2, 4}, {1, 1, 3, 3}, {1, 2, 1, 4}, {1, 2, 2, 3}}
{{1, 1, 2, 5}, {1, 1, 3, 4}, {1, 2, 1, 5}, {1, 2, 2, 4}, {1, 2, 3, 3}, ... 
{{1, 1, 3, 5}, {1, 2, 2, 5}, {1, 2, 3, 4}, {1, 3, 2, 4}}
{{1, 1, 3, 6}, {1, 2, 2, 6}, {1, 2, 4, 4}, {1, 2, 5, 3}, {1, 3, 2, 5}}
{{1, 2, 4, 5}, {1, 3, 2, 6}}
{{1, 2, 6, 4}, {1, 3, 2, 7}}
{{1, 1, 1, 4, 7}, {1, 1, 2, 3, 7}, {1, 1, 2, 5, 5}, {1, 1, 3, 2, 7}, ...
{{1, 1, 1, 4, 8}, {1, 1, 2, 3, 8}, {1, 1, 2, 5, 6}, {1, 1, 2, 6, 5}, ... 
{{1, 1, 3, 3, 8}, {1, 1, 3, 5, 6}, {1, 1, 4, 3, 7}, {1, 2, 1, 5, 7}, ... 
{{1, 1, 2, 8, 5}, {1, 1, 3, 3, 9}, {1, 1, 3, 6, 6}, {1, 1, 4, 3, 8}, ... 
{{1, 1, 3, 6, 7}, {1, 1, 4, 3, 9}, {1, 2, 3, 4, 8}, {1, 2, 5, 4, 6}, ...
{{1, 1, 4, 3, 10}, {1, 2, 2, 8, 6}, {1, 2, 4, 5, 7}, {1, 2, 6, 6, 4}, ...
{{1, 1, 1, 3, 4, 10}, {1, 1, 1, 3, 7, 7}, {1, 1, 1, 4, 3, 10}, ... 
{{1, 3, 10, 2, 5}} 
{{1, 1, 1, 4, 4, 11}, {1, 1, 1, 4, 7, 8}, {1, 1, 1, 5, 4, 10}, ... 
{{1, 1, 1, 4, 4, 12}, {1, 1, 1, 4, 8, 8}, {1, 1, 1, 5, 4, 11}, ... 
{{1, 1, 1, 4, 8, 9}, {1, 1, 1, 5, 4, 12}, {1, 1, 2, 8, 7, 5}, ... 
{{1, 1, 1, 5, 4, 13}, {1, 1, 2, 5, 6, 10}, {1, 1, 2, 8, 8, 5}, ... 
{{1, 1, 3, 4, 6, 11}, {1, 1, 3, 7, 6, 8}, {1, 1, 4, 4, 3, 13}, ... 
{{1, 1, 3, 8, 9, 5}, {1, 1, 4, 3, 7, 11}, {1, 1, 4, 4, 3, 14}, ... 
{{1, 3, 11, 5, 2, 6}} 
{{1, 1, 1, 1, 5, 5, 15}, {1, 1, 1, 1, 5, 10, 10}, {1, 1, 1, 1, 6, 5, 14}, ... 
{{1, 1, 1, 1, 5, 10, 11}, {1, 1, 1, 1, 6, 5, 15}, {1, 1, 1, 2, 6, 7, 12}, ... 
{{1, 2, 5, 4, 6, 13}, {1, 2, 7, 4, 12, 5}, {1, 3, 2, 7, 8, 10}, ... 

Would the difference basis partition for {1, 3, 11, 5, 2, 6} be {1, 4, 15, 20, 22, 28}? Or are these entirely different sequences that are identical for the first 92 terms?

[Christian Sievers]

Yes, but I'd use 0 instead of 28.

I also didn't know what a difference basis was, before I read the first sentence of the comment section, which defines:

A subset B is called a difference-basis for an Abelian group G if B-B=G.

So, each element must be expressible as the difference (in the group, so mod n) of two elements of B.

It's essentially the same, but an example that shows something else than what the name of the sequence describes, would be confusing.

BTW, your list misses n=3. Also, since the order matters, I'd rather call it composition instead of partition.

[Ed Pegg]

I'm used to the notation in the Singer paper, 
https://www.ams.org/journals/tran/1938-043-03/S0002-9947-1938-1501951-4/S0002-9947-1938-1501951-4.pdf  

Page 384

a(n) Perfect Difference Set          Perfect Partition

7      0 1 3                                       1 2 4  
21    0 1 4 14 16                             1 3 10 2 5  
73    0 1 3 7 15 31 36 54 63           1 2 4 8 16 5 18 9 10  
13    0 1 3 9                                    1 2 6 4
91    0 1 3 9 27 49 56 61 77 81      1 2 6 18 22 7 5 16 4 10 
31    0 1 3 8 12 18                          1 2 5 4 6 13

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[Ed Pegg]

I'm trying to solve a(111).  The prediction is that a 12 item basis / partition exists.  

With considerable effort, I've found 3492 13 item partitions for 111.  Here's a sampling:  

{{1,4,16,2,4,3,10,14,1,11,8,1,36}, {1,3,3,2,8,8,19,22,1,22,9,2,11},{1,8,14,4,13,28,3,2,2,6,15,4,11}, {1,5,7,3,6,32,18,6,2,2,7,8,14},{1,8,3,15,2,5,7,10,13,6,4,21,16}, {1,3,14,11,2,2,6,7,2,5,16,34,8}, {1,4,28,5,8,2,3,3,3,11,5,7,31}, {1,3,2,32,2,14,7,4,8,10,2,17,9}, {1,9,4,30,8,9,6,3,2,7,9,4,19}, {1,2,37,9,5,5,9,6,8,8,4,7,10}, {1,2,25,5,4,4,4,4,4,6,1,28,23}, {1,5,22,8,12,29,3,4,2,7,3,4,11}, {1,3,9,1,6,4,2,15,8,24,8,25,5}, {1,17,4,6,8,5,6,7,2,16,12,3,24}, {1,1,5,15,1,3,18,27,1,8,2,17,12}, {1,7,10,9,3,6,5,20,13,4,2,15,16}, {1,2,5,5,9,17,20,7,5,6,19,4,11}, {1,2,2,1,6,6,10,10,10,4,7,8,44}, {1,3,5,18,12,2,17,11,14,8,13,1,6}, {1,9,2,3,10,6,17,3,4,4,30,5,17}}

I'm running out of tweaking ideas to generate more.  Theoretically, if   a(111)=12, there will be 12 alternate solutions, and about 1200 ways to bring these to 13 parts. Various surgery methods can be used to alter these so that they still provide a covering.  So, if you have enough 13 part solutions, eventually this set and the solution set will overlap, and one of the items will be reducible to a 12 part solution. It worked really well for Sparse Rulers.

If anyone would like to play around with the  3492 13 item partitions for 111, let me know and I'll email you a copy.

[David desJardins]

I haven’t tried, but isn’t it also true that if multiply by any X coprime to 111 = 3*37, you get a related solution? So, that’s more than 12 related solutions in a family, it’s probably 12*2*36 = 864. Unless there are symmetries, which would also be helpful.

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[Christian Sievers]

? b=[0,10,17,33,38,39,41,53,73,84,98,102];
? #b
%2 = 12
? #setbinop((x,y)->(x-y)%111,b,b)
%3 = 111

[Ed Pegg]

Here are 387 12-partitions of 111 that each cover all but one pair of cyclic sums. For example, the last partition,  {2,7,36,4,3,7,8,16,5,6,6,11} cannot cover cyclic sums of 1 or 110. The only pair not missed in the whole set is 37, 74.

I'm starting to think there isn't a perfect 12-partition of 111, but I'm still running programs. 

{{1,1,2,5,11,26,2,23,6,21,3,10}, {1,1,2,6,5,3,25,12,7,27,2,20}, {1,1,2,7,6,14,5,12,16,8,18,21}, {1,1,2,11,4,5,17,12,16,11,25,6}, {1,1,2,15,11,29,5,4,23,8,6,6}, {1,1,3,3,4,16,20,9,22,2,17,13}, {1,1,3,4,22,25,7,14,6,17,1,10}, {1,1,3,11,14,7,5,8,35,3,6,17}, {1,1,3,18,8,6,11,2,7,16,28,10}, {1,1,4,1,4,25,8,20,8,24,3,12}, {1,1,4,15,24,18,11,3,9,10,8,7}, {1,1,4,17,11,27,8,3,9,7,10,13}, {1,1,5,3,9,5,28,13,12,4,11,19}, {1,1,5,8,32,12,4,19,2,1,17,9}, {1,1,5,12,3,31,4,7,25,1,13,8}, {1,1,6,3,15,22,6,23,4,12,5,13}, {1,1,6,10,11,4,20,10,9,3,23,13}, {1,1,6,16,4,30,9,5,3,15,10,11}, {1,1,8,24,21,10,4,3,12,1,5,21}, {1,1,8,40,2,3,20,3,4,11,6,12}, {1,1,9,5,44,5,7,5,3,4,6,21}, {1,1,12,6,11,10,5,4,3,35,9,14}, {1,1,17,16,27,5,4,3,8,2,4,23}, {1,1,18,14,26,5,4,5,3,7,6,21}, {1,2,2,4,5,7,10,6,31,4,15,24}, {1,2,2,15,1,7,6,24,2,9,1,41}, {1,2,2,16,19,6,8,22,12,13,4,6}, {1,2,2,22,20,13,8,9,6,10,7,11}, {1,2,3,4,8,14,3,13,11,12,21,19}, {1,2,3,4,8,14,11,16,8,11,13,20}, {1,2,3,7,4,8,9,15,5,18,12,27}, {1,2,3,11,4,4,9,10,26,7,5,29}, {1,2,3,11,4,25,8,22,9,3,16,7}, {1,2,3,11,9,15,4,8,10,20,13,15}, {1,2,4,5,21,27,9,4,6,10,8,14}, {1,2,4,8,19,1,14,9,17,21,5,10}, {1,2,4,9,5,9,10,11,1,34,8,17}, {1,2,4,10,7,13,13,5,17,12,19,8}, {1,2,4,11,9,14,8,12,5,16,19,10}, {1,2,4,12,8,30,4,5,23,8,4,10}, {1,2,4,13,8,3,9,14,10,5,27,15}, {1,2,4,17,5,9,16,8,10,10,17,12}, {1,2,4,18,10,12,14,5,8,17,12,8}, {1,2,4,18,13,11,8,25,9,5,12,3}, {1,2,4,20,34,5,5,11,7,10,4,8}, {1,2,4,25,9,17,11,7,12,3,13,7}, {1,2,5,4,14,4,9,20,15,6,15,16}, {1,2,5,6,5,4,27,17,12,12,13,7}, {1,2,5,9,21,6,13,5,24,10,12,3}, {1,2,5,10,9,24,4,12,13,11,6,14}, {1,2,5,12,1,15,10,4,18,9,31,3}, {1,2,5,16,1,13,6,10,18,27,4,8}, {1,2,6,3,19,5,7,30,15,6,4,13}, {1,2,6,4,5,20,16,14,12,21,6,4}, {1,2,6,4,15,2,14,4,2,5,33,23}, {1,2,6,5,12,9,3,15,4,27,20,7}, {1,2,6,5,14,7,4,11,1,8,9,43}, {1,2,6,7,17,5,5,18,3,36,8,3}, {1,2,6,7,22,19,10,17,4,12,9,2}, {1,2,6,9,13,7,5,7,4,10,6,41}, {1,2,6,10,1,4,7,14,13,16,5,32}, {1,2,6,10,41,4,13,4,11,12,2,5}, {1,2,6,12,4,10,5,27,11,6,7,20}, {1,2,6,17,7,5,15,27,18,3,7,3}, {1,2,6,22,4,2,13,10,16,17,7,11}, {1,2,6,22,26,4,7,5,20,4,10,4}, {1,2,7,1,14,28,21,2,3,1,12,19}, {1,2,7,4,13,20,13,12,11,7,16,5}, {1,2,7,9,12,8,13,14,12,11,17,5}, {1,2,7,13,4,16,14,5,6,7,8,28}, {1,2,7,15,30,8,7,5,6,17,4,9}, {1,2,8,3,17,1,26,15,4,5,7,22}, {1,2,8,5,9,11,7,19,4,17,17,11}, {1,2,8,9,18,8,13,19,6,16,7,4}, {1,2,8,15,4,1,13,21,24,10,6,6}, {1,2,8,20,8,13,27,10,7,7,5,3}, {1,2,8,23,24,8,6,13,1,16,5,4}, {1,2,9,5,6,7,15,6,4,33,4,19}, {1,2,9,8,10,4,33,5,10,6,7,16}, {1,2,9,11,9,4,7,14,5,34,10,5}, {1,2,10,4,9,21,1,5,32,11,8,7}, {1,2,10,6,7,20,16,11,4,5,8,21}, {1,2,10,7,4,6,20,5,9,24,8,15}, {1,2,10,8,10,4,5,6,16,26,7,16}, {1,2,10,14,9,21,7,2,25,4,11,5}, {1,2,10,15,5,9,8,18,22,11,7,3}, {1,2,10,17,5,9,5,2,4,33,8,15}, {1,2,10,18,8,8,25,15,5,9,6,4}, {1,2,10,22,7,1,17,28,5,4,11,3}, {1,2,11,3,1,6,12,8,5,25,9,28}, {1,2,11,21,1,5,4,8,7,23,26,2}, {1,2,11,32,3,2,21,11,4,8,10,6}, {1,2,11,35,6,12,7,8,4,5,5,15}, {1,2,12,7,10,1,5,4,16,4,4,45}, {1,2,12,9,8,16,2,20,1,30,6,4}, {1,2,12,10,8,5,4,17,7,29,11,5}, {1,2,12,11,10,13,9,9,20,8,12,4}, {1,2,12,19,21,10,4,6,7,5,16,8}, {1,2,13,18,2,3,5,4,7,6,24,26}, {1,2,13,22,7,18,8,20,4,2,5,9}, {1,2,14,4,7,35,10,2,7,6,18,5}, {1,2,14,4,25,8,12,7,9,6,13,10}, {1,2,14,6,11,4,7,8,5,5,39,9}, {1,2,14,19,8,5,7,11,19,4,6,15}, {1,2,15,4,10,4,12,8,3,2,44,6}, {1,2,15,5,4,16,6,33,11,7,1,10}, {1,2,15,9,10,4,2,6,5,8,20,29}, {1,2,15,19,12,4,24,5,6,3,13,7}, {1,2,16,5,22,15,5,7,24,4,4,6}, {1,2,16,14,17,11,10,14,14,6,2,4}, {1,2,17,14,11,10,16,13,9,6,8,4}, {1,2,18,31,8,1,9,14,11,4,7,5}, {1,2,19,6,2,4,3,17,39,5,5,8}, {1,2,20,8,16,13,16,17,7,3,5,3}, {1,2,20,9,2,5,8,6,4,17,28,9}, {1,2,20,10,7,14,5,6,18,16,4,8}, {1,2,21,12,31,6,5,4,3,2,8,16}, {1,2,22,9,10,17,11,19,2,6,8,4}, {1,2,22,9,20,8,4,6,5,4,17,13}, {1,2,24,4,10,5,15,17,12,6,7,8}, {1,2,24,8,6,3,10,8,4,11,5,29}, {1,2,25,4,4,3,13,6,21,18,5,9}, {1,2,25,31,1,8,7,14,5,7,7,3}, {1,2,28,15,7,22,4,8,1,5,11,7}, {1,2,34,6,5,4,4,10,12,9,7,17}, {1,3,1,3,2,43,12,14,2,11,2,17}, {1,3,1,3,5,29,2,15,30,2,14,6}, {1,3,2,2,12,12,16,11,22,13,8,9}, {1,3,2,5,13,3,14,8,22,12,19,9}, {1,3,2,7,11,10,5,6,8,8,9,41}, {1,3,2,8,5,19,7,9,11,22,7,17}, {1,3,2,8,10,25,7,2,26,11,1,15}, {1,3,2,13,7,10,11,2,29,9,16,8}, {1,3,2,14,15,11,6,1,23,14,13,8}, {1,3,2,27,8,10,2,20,14,9,8,7}, {1,3,2,44,3,2,9,10,8,7,6,16}, {1,3,3,2,12,13,10,19,16,11,4,17}, {1,3,4,2,17,5,14,20,1,11,18,15}, {1,3,5,1,32,7,19,1,17,11,2,12}, {1,3,5,3,13,12,2,20,20,9,17,6}, {1,3,5,4,7,17,22,2,19,6,15,10}, {1,3,5,6,7,11,6,25,2,10,16,19}, {1,3,5,6,7,17,16,10,2,17,8,19}, {1,3,5,17,13,2,10,6,27,7,10,10}, {1,3,5,27,13,7,14,2,10,5,6,18}, {1,3,5,30,2,12,16,17,3,7,11,4}, {1,3,6,2,5,16,12,10,20,19,3,14}, {1,3,6,2,6,2,5,29,4,22,4,27}, {1,3,6,3,2,6,16,2,23,7,12,30}, {1,3,6,5,2,18,8,24,8,24,6,6}, {1,3,6,5,15,17,2,14,27,1,12,8}, {1,3,6,14,8,7,19,11,2,23,5,12}, {1,3,7,8,14,9,12,13,17,20,2,5}, {1,3,7,27,12,2,24,7,9,5,8,6}, {1,3,8,2,4,5,20,7,21,7,21,12}, {1,3,8,2,15,7,9,5,6,33,4,18}, {1,3,8,7,11,38,2,8,6,7,15,5}, {1,3,8,9,6,23,2,5,14,18,10,12}, {1,3,9,2,4,22,5,22,8,16,13,6}, {1,3,9,2,23,5,15,2,16,8,21,6}, {1,3,9,3,27,14,18,1,5,21,2,7}, {1,3,9,11,3,22,7,5,33,2,6,9}, {1,3,9,11,6,8,2,5,20,8,20,18}, {1,3,11,6,15,19,6,12,26,3,2,7}, {1,3,11,6,34,9,10,6,2,5,3,21}, {1,3,11,7,5,4,26,6,2,17,20,9}, {1,3,11,10,6,35,3,10,5,2,17,8}, {1,3,11,13,2,6,10,2,5,2,34,22}, {1,3,12,9,23,4,33,2,6,5,7,6}, {1,3,13,2,3,9,28,6,14,10,15,7}, {1,3,13,2,6,4,26,7,7,11,9,22}, {1,3,13,2,6,7,5,10,30,9,2,23}, {1,3,13,9,2,27,7,6,6,23,10,4}, {1,3,13,12,20,2,20,6,11,9,10,4}, {1,3,14,2,10,11,39,7,1,5,3,15}, {1,3,14,6,7,21,5,11,25,4,6,8}, {1,3,15,7,2,3,11,6,33,13,8,9}, {1,3,16,5,10,7,6,23,3,27,2,8}, {1,3,18,5,9,15,2,14,25,12,1,6}, {1,3,18,20,9,2,19,13,3,9,8,6}, {1,3,20,1,14,18,8,2,17,15,7,5}, {1,3,20,12,2,19,8,18,13,4,5,6}, {1,3,24,5,21,1,8,11,23,2,6,6}, {1,3,25,6,6,20,16,11,2,7,10,4}, {1,3,31,11,8,6,7,5,10,2,11,16}, {1,3,33,15,7,3,9,9,2,16,8,5}, {1,4,2,3,13,8,7,14,11,9,27,12}, {1,4,2,4,3,19,27,15,2,10,8,16}, {1,4,2,8,11,3,13,19,12,5,18,15}, {1,4,2,9,2,22,8,14,28,3,7,11}, {1,4,2,10,17,5,3,10,16,20,9,14}, {1,4,2,15,23,3,6,28,5,11,8,5}, {1,4,2,25,15,7,19,9,13,3,8,5}, {1,4,2,32,1,18,8,3,12,13,8,9}, {1,4,3,10,2,9,10,4,10,16,6,36}, {1,4,3,11,3,9,36,10,2,4,9,19}, {1,4,4,24,11,13,13,16,3,15,2,5}, {1,4,5,1,9,30,3,22,13,2,14,7}, {1,4,5,8,8,23,20,2,12,3,19,6}, {1,4,6,15,7,9,3,15,2,6,30,13}, {1,4,6,22,27,8,4,3,2,14,7,13}, {1,4,7,2,6,14,19,14,18,3,7,16}, {1,4,7,8,16,7,2,39,7,3,3,14}, {1,4,7,9,20,14,8,10,13,2,17,6}, {1,4,7,14,20,10,3,6,23,6,2,15}, {1,4,7,15,32,3,6,2,8,20,1,12}, {1,4,7,16,2,17,6,3,11,13,21,10}, {1,4,8,2,7,17,11,14,16,13,13,5}, {1,4,8,3,32,9,10,7,12,2,6,17}, {1,4,8,6,11,4,20,3,13,32,2,7}, {1,4,8,12,6,21,2,15,1,32,3,6}, {1,4,8,14,7,9,2,17,3,3,33,10}, {1,4,8,21,23,9,3,4,2,11,11,14}, {1,4,9,7,12,3,11,30,10,2,6,16}, {1,4,10,17,12,5,3,3,13,2,7,34}, {1,4,10,21,2,6,11,7,9,13,12,15}, {1,4,12,19,6,3,11,27,5,2,8,13}, {1,4,15,2,2,4,10,13,9,3,42,6}, {1,4,15,18,7,15,9,3,23,6,2,8}, {1,4,16,2,7,1,17,6,7,36,3,11}, {1,4,16,9,31,2,24,2,5,3,3,11}, {1,4,17,25,3,12,11,2,7,16,8,5}, {1,4,18,14,28,1,6,6,3,8,2,20}, {1,4,20,1,13,2,15,27,10,9,3,6}, {1,4,24,24,2,10,9,4,16,3,8,6}, {1,4,27,18,3,4,10,6,6,22,2,8}, {1,4,37,11,2,17,12,4,3,7,8,5}, {1,5,2,4,12,31,3,14,10,9,6,14}, {1,5,2,7,3,16,11,21,4,18,18,5}, {1,5,2,12,21,4,9,15,3,23,6,10}, {1,5,2,14,30,5,15,3,6,4,15,11}, {1,5,3,17,5,5,2,11,29,14,4,15}, {1,5,3,27,2,17,13,10,8,10,4,11}, {1,5,4,5,2,30,8,20,3,8,13,12}, {1,5,4,7,24,21,5,7,1,14,3,19}, {1,5,4,11,5,8,31,10,12,7,3,14}, {1,5,4,13,3,8,19,12,11,7,14,14}, {1,5,4,18,3,8,5,2,5,12,2,46}, {1,5,6,3,4,21,2,8,14,16,17,14}, {1,5,8,10,4,3,9,3,36,2,9,21}, {1,5,8,12,3,4,10,4,31,9,2,22}, {1,5,9,3,7,16,11,2,2,6,22,27}, {1,5,10,2,2,7,31,22,3,5,5,18}, {1,5,10,3,31,9,11,4,8,2,7,20}, {1,5,10,6,8,10,7,2,11,12,3,36}, {1,5,11,3,5,10,12,28,4,7,2,23}, {1,5,14,3,9,2,16,8,7,6,4,36}, {1,5,14,8,10,3,4,9,2,31,1,23}, {1,5,14,13,16,5,32,3,4,2,6,10}, {1,5,14,20,1,16,27,2,3,4,8,10}, {1,5,15,4,20,19,3,9,2,16,10,7}, {1,5,16,4,3,37,2,8,9,2,12,12}, {1,5,21,28,10,2,3,10,7,12,4,8}, {1,5,22,3,8,2,2,14,7,9,20,18}, {1,5,22,10,2,17,24,5,4,11,3,7}, {1,5,25,4,4,8,20,23,3,9,2,7}, {1,6,2,3,16,13,9,13,4,20,10,14}, {1,6,2,4,16,25,15,5,11,3,14,9}, {1,6,3,13,2,19,12,15,5,6,22,7}, {1,6,5,4,2,6,16,3,10,10,34,14}, {1,6,5,16,20,14,4,13,2,7,3,20}, {1,6,6,3,8,24,5,10,4,24,2,18}, {1,6,6,4,15,33,1,8,9,2,3,23}, {1,6,7,3,18,2,3,21,11,27,4,8}, {1,6,8,4,12,38,4,2,3,13,10,10}, {1,6,8,10,9,2,2,41,5,8,4,15}, {1,6,9,32,9,5,4,22,3,8,2,10}, {1,6,9,39,11,5,3,2,8,2,2,23}, {1,6,10,2,3,11,9,4,28,6,2,29}, {1,6,11,3,5,4,4,30,13,2,22,10}, {1,6,14,3,7,5,11,2,9,41,4,8}, {1,6,14,4,8,1,10,17,2,3,31,14}, {1,6,15,8,20,4,6,25,9,2,3,12}, {1,6,16,11,9,2,3,10,4,4,8,37}, {1,6,17,2,3,4,7,36,5,10,8,12}, {1,6,17,9,10,4,6,2,13,5,11,27}, {1,6,18,4,32,11,2,3,7,7,12,8}, {1,6,20,5,2,3,6,8,4,17,24,15}, {1,6,28,15,3,2,12,4,8,1,10,21}, {1,6,29,3,22,11,10,2,10,4,5,8}, {1,7,2,2,26,17,3,14,5,13,6,15}, {1,7,2,26,14,5,23,4,6,8,3,12}, {1,7,3,2,2,17,1,34,2,4,23,15}, {1,7,3,11,3,2,28,6,9,29,2,10}, {1,7,3,15,20,23,2,4,6,17,5,8}, {1,7,3,23,4,5,9,6,25,15,2,11}, {1,7,4,2,3,21,11,18,5,5,15,19}, {1,7,7,6,4,19,3,9,2,16,24,13}, {1,7,8,4,6,18,3,2,9,2,22,29}, {1,7,9,3,2,4,4,31,1,23,15,11}, {1,7,9,9,4,6,3,2,21,23,14,12}, {1,7,10,2,5,23,15,13,3,6,5,21}, {1,7,10,2,28,6,9,5,11,10,3,19}, {1,7,10,30,4,2,13,7,2,3,11,21}, {1,7,11,12,26,1,16,3,2,4,15,13}, {1,7,12,2,4,5,10,14,3,10,35,8}, {1,7,12,4,2,3,5,12,15,27,13,10}, {1,7,12,6,4,5,23,9,2,26,3,13}, {1,7,13,3,2,4,10,17,15,2,26,11}, {1,7,13,4,34,2,3,6,12,10,4,15}, {1,7,14,6,4,2,3,10,30,11,7,16}, {1,7,16,9,22,18,4,6,15,2,3,8}, {1,7,16,17,14,7,2,3,6,4,15,19}, {1,7,18,18,3,11,2,24,5,4,6,12}, {1,7,19,3,2,9,7,28,6,4,13,12}, {1,7,22,6,4,2,3,11,5,27,6,17}, {1,7,28,6,10,1,19,13,3,2,7,14}, {1,7,51,3,2,3,3,12,4,12,4,9}, {1,8,2,3,12,7,6,21,2,14,4,31}, {1,8,3,4,2,10,4,24,22,8,5,20}, {1,8,3,4,11,3,10,6,17,5,20,23}, {1,8,3,6,16,23,4,21,10,5,2,12}, {1,8,3,39,7,6,11,4,10,2,3,17}, {1,8,4,3,18,11,6,20,2,24,5,9}, {1,8,4,6,16,17,11,3,17,5,2,21}, {1,8,5,2,4,30,7,10,12,6,3,23}, {1,8,5,4,7,3,12,21,2,20,12,16}, {1,8,5,7,3,27,11,6,12,4,2,25}, {1,8,6,5,17,7,3,21,2,16,12,13}, {1,8,6,7,4,3,11,5,32,10,12,12}, {1,8,7,4,2,22,3,14,17,5,18,10}, {1,8,10,5,3,14,4,7,6,16,26,11}, {1,8,10,5,7,10,4,2,11,3,26,24}, {1,8,14,3,2,13,13,7,29,6,4,11}, {1,8,16,5,2,4,6,3,19,18,3,26}, {1,8,21,6,12,2,2,3,12,11,10,23}, {1,8,22,15,3,2,7,19,4,6,11,13}, {1,9,5,3,4,16,6,2,11,13,1,40}, {1,9,5,3,9,13,7,4,2,19,23,16}, {1,9,7,13,7,5,3,11,32,2,4,17}, {1,9,8,4,12,32,3,3,20,5,2,12}, {1,9,11,4,7,20,6,23,12,3,2,13}, {1,9,24,2,3,3,12,7,4,17,16,13}, {1,10,2,8,7,9,4,14,5,26,3,22}, {1,10,3,10,5,2,2,42,12,8,6,10}, {1,10,3,16,2,4,5,34,9,8,7,12}, {1,10,5,13,3,9,8,2,4,20,7,29}, {1,10,6,8,7,2,5,6,36,4,8,18}, {1,10,8,3,19,4,2,2,5,7,17,33}, {1,10,22,8,7,4,2,3,20,6,12,16}, {1,11,2,2,5,21,19,8,19,3,3,17}, {1,11,3,4,6,26,17,14,2,5,2,20}, {1,11,4,2,3,8,14,10,14,26,7,11}, {1,11,5,9,4,2,2,15,24,3,7,28}, {1,11,8,25,13,3,2,9,4,6,7,22}, {1,11,12,2,4,13,3,30,8,2,5,20}, {1,11,14,3,3,10,8,10,5,2,2,42}, {1,11,37,2,4,4,8,7,3,5,9,20}, {1,12,2,8,10,7,4,5,16,3,6,37}, {1,12,3,5,19,2,5,4,6,8,14,32}, {1,12,4,3,18,17,22,5,4,2,8,15}, {1,12,8,4,5,6,14,2,16,3,7,33}, {1,12,8,16,7,4,3,2,13,4,6,35}, {1,12,12,2,6,12,3,7,4,5,31,16}, {1,13,8,4,6,3,2,14,20,7,17,16}, {1,13,18,3,2,6,4,7,9,10,25,13}, {1,13,18,3,5,3,4,5,10,6,20,23}, {1,13,20,8,16,2,4,5,3,7,10,22}, {1,13,21,2,3,4,8,10,6,14,11,18}, {1,14,4,2,7,3,23,11,5,17,7,17}, {1,14,6,2,3,7,4,13,18,9,19,15}, {1,14,8,4,5,2,18,10,3,3,18,25}, {1,14,10,20,7,6,3,2,17,4,8,19}, {1,14,25,7,10,2,4,3,11,8,5,21}, {1,15,3,4,5,2,6,25,11,15,6,18}, {1,15,3,28,23,4,6,2,5,4,5,15}, {1,15,30,5,7,2,3,8,13,6,4,17}, {1,16,2,4,5,3,7,16,13,20,1,23}, {1,16,3,22,2,7,2,4,8,18,10,18}, {1,16,6,4,3,2,9,10,15,8,12,25}, {1,16,9,4,2,18,3,7,4,3,5,39}, {1,16,28,2,4,3,5,10,3,8,12,19}, {1,17,2,10,8,2,5,6,3,28,4,25}, {1,17,15,7,3,5,4,2,13,7,9,28}, {1,17,23,7,4,2,3,10,12,4,8,20}, {1,18,10,21,15,6,2,5,4,3,2,24}, {1,19,2,9,5,24,8,4,6,7,3,23}, {1,19,4,2,6,5,9,7,15,3,10,30}, {1,22,3,9,2,5,2,6,4,28,1,28}, {2,3,4,2,7,3,14,8,22,15,13,18}, {2,3,7,5,6,5,22,10,9,30,4,8}, {2,3,7,7,9,6,15,11,18,21,4,8}, {2,3,11,7,4,13,12,8,10,9,6,26}, {2,4,17,12,10,3,5,6,4,9,7,32}, {2,6,24,18,9,4,12,10,11,3,5,7}, {2,7,4,5,4,6,13,8,3,14,12,33}, {2,7,36,4,3,7,8,16,5,6,6,11}}

[Christian Sievers]

I guess my last post was too terse (or "perfect" poses an extra condition that I'm missing), but there I showed a difference base of size 12 for n=111.

In your notation it would be this:

{10,7,16,5,1,2,12,20,11,14,4,9}

(I just got lucky running a trivial rendering of the problem in an ASP solver for about an hour.)

[Ed Pegg]

I need to get an ASP solver.   Any chance you could run for 106, 109, 110?  Those are the current gaps.  And maybe 112, 113, 114.  

{(*101*){1, 14, 10, 20, 7, 6, 3, 2, 17, 4, 8, 9},  
 (*102*){1, 3, 12, 3, 21, 2, 8, 9, 5, 6, 7, 25},  
 (*103*){1, 2, 14, 12, 2, 19, 6, 5, 4, 18, 13, 7},  
 (*104*){1, 2, 12, 2, 25, 4, 9, 10, 7, 11, 16, 5},  
 (*105*){1, 2, 9, 8, 14, 4, 15, 7, 6, 10, 5, 24},  
 (*107*){1, 8, 10, 5, 7, 21, 4, 2, 11, 3, 26, 9},  
 (*108*){1, 2, 12, 6, 25, 4, 9, 10, 7, 11, 16, 5},  
 (*111*){1, 2, 12, 20, 11, 14, 4, 9, 10, 7, 16, 5}}    

Curiously, your solution has 7 multiplicative variants.  

{{1,2,12,20,11,14,4,9,10,7,16,5}, {1,3,4,5,6,17,10,14,2,28,9,12},{1,3,8,13,2,7,23,14,5,19,10,6}, {1,4,8,6,9,2,5,26,10,3,18,19}, {1,4,16,3,5,6,12,13,2,7,10,32}, {1,14,20,5,13,8,3,4,2,10,9,22},{1,18,12,16,5,11,6,3,4,8,2,25}}

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[David desJardins]

On Wed, Jul 23, 2025 at 10:58 PM Ed Pegg <edp...@gmail.com> wrote:
>
> Curiously, your solution has 7 multiplicative variants.

What? As I posted before, his solution gives 864 = 12*2*36 equivalent solutions. Shift any element to position 0, and then multiply by any value coprime to 111. I even checked that they are all distinct.

{0,1,2,5,12,27,36,38,44,52,65,93}
{0,1,2,5,17,34,40,48,62,85,92,103}
{0,1,2,5,25,35,42,46,82,85,98,104}
{0,1,2,9,15,28,31,67,71,78,88,108}
{0,1,2,10,21,28,51,65,73,79,96,108}
{0,1,2,20,48,61,69,75,77,86,101,108}
{0,1,3,10,27,59,70,73,77,81,93,106}
{0,1,3,15,35,46,60,64,73,83,90,106}
{0,1,3,19,28,34,42,54,64,90,101,107}
{0,1,3,29,48,60,64,73,74,84,91,106}
{0,1,4,8,13,19,36,46,60,62,90,99}
{0,1,4,8,38,40,51,70,84,93,94,99}
{0,1,4,9,20,22,29,35,52,62,74,76}
{0,1,4,11,26,35,37,43,51,64,92,110}
{0,1,4,12,25,27,34,57,71,76,95,105}
{0,1,4,15,20,25,27,33,57,67,76,98}
{0,1,4,16,21,43,45,53,81,92,99,106}
{0,1,4,16,33,39,47,61,84,91,102,110}
{0,1,4,18,30,64,71,73,79,84,89,108}
{0,1,4,18,33,38,39,49,61,85,102,104}
{0,1,4,21,28,36,43,81,83,93,95,106}
{0,1,4,22,52,57,66,72,74,76,83,99}
{0,1,4,23,28,33,39,41,48,82,94,108}
{0,1,4,24,34,41,45,81,84,97,103,110}
{0,1,5,7,17,43,64,67,75,83,89,98}
{0,1,5,8,18,30,61,64,79,84,103,105}
{0,1,5,11,22,48,58,70,78,84,93,109}
{0,1,5,13,19,28,30,35,61,71,74,92}
{0,1,5,14,48,54,56,66,81,88,92,95}
{0,1,5,19,28,34,36,42,80,87,90,100}
{0,1,5,20,22,32,39,62,87,90,96,98}
{0,1,5,21,24,29,35,47,60,62,69,79}
{0,1,6,7,15,17,29,33,68,71,81,92}
{0,1,6,7,26,37,47,50,85,89,101,103}
{0,1,6,13,20,31,59,67,69,91,96,108}
{0,1,6,15,18,22,48,50,58,61,88,92}
{0,1,6,17,19,29,31,69,76,84,91,108}
{0,1,6,18,19,22,26,56,58,69,88,102}
{0,1,6,19,31,35,39,42,53,85,102,109}
{0,1,6,21,28,38,39,48,52,64,83,109}
{0,1,6,21,35,38,39,46,48,65,89,101}
{0,1,6,22,29,39,48,52,66,77,97,109}
{0,1,7,9,28,33,48,51,82,94,104,107}
{0,1,7,10,12,23,27,50,58,79,87,93}
{0,1,7,12,27,28,36,38,76,79,89,93}
{0,1,7,17,36,41,55,78,85,87,100,108}
{0,1,8,10,27,51,63,73,74,79,94,108}
{0,1,8,13,27,29,31,37,77,80,86,97}
{0,1,8,14,27,30,66,70,77,87,107,110}
{0,1,8,17,29,54,65,67,70,79,85,89}
{0,1,9,11,23,27,62,65,75,86,105,106}
{0,1,9,11,49,52,62,66,84,85,91,96}
{0,1,9,20,27,50,64,72,78,95,107,110}
{0,1,9,50,53,55,69,72,76,82,87,100}
{0,1,10,14,26,45,71,73,74,79,94,101}
{0,1,10,24,43,54,56,86,90,93,94,106}
{0,1,11,18,33,38,39,41,67,86,98,102}
{0,1,11,23,47,64,66,73,74,77,91,106}
{0,1,12,22,25,32,70,76,78,84,93,107}
{0,1,12,25,30,36,40,43,57,59,62,103}
{0,1,13,18,19,28,42,61,72,74,104,108}
{0,1,13,22,50,52,66,76,93,99,104,108}
{0,1,13,29,36,38,40,46,55,60,90,108}
{0,1,13,31,52,63,67,69,72,77,89,96}
{0,1,14,16,22,25,50,73,80,90,92,107}
{0,1,14,23,29,37,45,48,69,95,105,107}
{0,1,14,36,45,55,79,85,87,92,97,108}
{0,1,14,39,44,54,56,60,63,71,89,91}
{0,1,15,26,32,35,75,81,83,85,99,104}
{0,1,15,35,40,53,61,64,68,70,80,89}
{0,1,16,18,20,26,29,50,75,84,89,96}
{0,1,16,21,27,28,46,50,60,63,101,103}
{0,1,16,23,28,37,62,83,86,92,94,96}
{0,1,16,23,35,40,43,45,49,60,81,99}
{0,1,17,20,24,31,46,56,58,64,98,107}
{0,1,19,23,33,36,74,76,84,85,100,105}
{0,1,19,25,33,54,62,85,89,100,102,105}
{0,1,19,31,47,52,63,69,72,76,84,86}
{0,1,19,47,60,68,74,76,85,100,107,110}
{0,1,20,24,51,54,62,64,90,94,97,106}
{0,1,20,31,41,44,79,83,95,97,105,106}
{0,1,20,38,41,51,77,82,84,93,99,107}
{0,1,21,23,41,49,52,56,58,68,73,98}
{0,1,23,27,33,42,45,47,58,83,95,104}
{0,1,23,32,42,44,48,51,59,72,77,97}
{0,1,26,28,36,40,43,49,60,65,81,93}
{0,1,33,43,50,52,65,77,83,88,91,107}
{0,1,36,38,50,60,77,83,90,92,103,108}
{0,2,3,7,13,24,50,60,72,80,86,95}
{0,2,3,8,21,33,37,41,44,55,87,104}
{0,2,3,8,23,30,40,41,50,54,66,85}
{0,2,3,8,24,31,41,50,54,68,79,99}
{0,2,4,10,13,34,59,68,73,80,95,96}
{0,2,4,10,19,24,54,72,75,76,88,104}
{0,2,4,10,50,53,59,70,84,85,92,97}
{0,2,4,11,27,39,40,43,61,91,96,105}
{0,2,4,18,23,30,31,45,56,62,65,105}
{0,2,4,19,20,35,42,47,56,81,102,105}
{0,2,5,10,22,29,44,45,57,75,96,107}
{0,2,5,11,12,30,36,44,65,73,96,100}
{0,2,5,14,20,24,46,47,54,63,75,100}
{0,2,5,46,54,55,66,79,84,90,94,97}
{0,2,6,7,20,29,35,43,51,54,75,101}
{0,2,6,9,17,30,35,55,69,70,92,101}
{0,2,6,9,17,35,37,57,58,71,96,101}
{0,2,6,17,38,56,68,69,84,91,103,108}
{0,2,7,12,23,26,27,40,62,71,81,105}
{0,2,7,33,43,46,64,83,84,88,96,102}
{0,2,8,9,13,16,26,38,69,72,87,92}
{0,2,8,11,32,57,66,71,78,93,94,109}
{0,2,8,11,36,59,66,76,78,93,97,98}
{0,2,8,13,18,37,40,41,44,58,70,104}
{0,2,8,16,29,57,75,76,77,80,87,102}
{0,2,8,17,22,52,70,73,74,86,102,109}
{0,2,8,17,31,35,36,47,57,60,67,105}
{0,2,8,32,42,51,73,86,87,90,101,106}
{0,2,8,42,51,55,56,72,75,79,86,101}
{0,2,8,46,53,56,66,77,78,82,96,105}
{0,2,8,48,51,57,68,82,83,90,95,109}
{0,2,9,10,13,27,42,47,48,58,70,94}
{0,2,9,15,32,42,54,56,91,92,95,100}
{0,2,9,19,51,52,56,72,75,80,86,98}
{0,2,9,25,37,38,41,59,89,94,103,109}
{0,2,9,26,58,69,72,76,80,92,105,110}
{0,2,9,32,46,51,70,80,86,87,90,98}
{0,2,9,43,55,69,72,73,76,95,100,105}
{0,2,10,11,16,17,36,47,57,60,95,99}
{0,2,10,11,26,31,37,38,56,60,70,73}
{0,2,10,13,40,44,63,64,69,78,81,85}
{0,2,10,14,17,23,34,39,55,67,85,86}
{0,2,10,38,49,56,63,68,69,72,84,89}
{0,2,11,17,25,29,30,49,67,70,80,106}
{0,2,11,26,33,36,37,38,56,84,97,105}
{0,2,12,14,25,30,31,34,51,58,66,73}
{0,2,12,14,52,59,67,74,91,94,95,100}
{0,2,12,17,42,55,56,76,78,96,104,107}
{0,2,12,19,42,67,70,76,78,91,92,96}
{0,2,12,21,43,44,58,78,83,96,104,107}
{0,2,12,27,34,38,41,57,58,62,71,105}
{0,2,12,38,59,62,70,78,84,93,106,107}
{0,2,13,17,40,48,69,77,83,101,102,108}
{0,2,13,18,19,22,39,46,54,61,99,101}
{0,2,13,18,21,22,57,59,71,81,98,104}
{0,2,13,32,46,55,56,61,73,74,77,81}
{0,2,13,38,50,59,66,67,89,93,99,108}
{0,2,14,18,53,56,66,77,96,97,102,103}
{0,2,14,24,41,47,54,56,67,72,75,76}
{0,2,14,34,45,59,63,72,82,89,105,110}
{0,2,15,16,20,35,37,47,54,77,102,105}
{0,2,15,23,26,27,33,43,62,67,81,104}
{0,2,15,27,33,38,41,57,61,62,94,104}
{0,2,16,19,23,29,34,47,58,59,67,108}
{0,2,16,21,28,29,43,54,60,63,103,109}
{0,2,16,26,43,49,54,58,61,62,74,83}
{0,2,17,18,33,40,45,54,79,100,103,109}
{0,2,17,21,22,35,37,43,46,71,94,101}
{0,2,18,27,33,41,53,63,89,100,106,110}
{0,2,19,43,55,65,66,71,86,100,103,104}
{0,2,20,28,31,35,37,47,52,77,90,91}
{0,2,21,26,41,44,75,87,97,100,104,105}
{0,2,22,23,36,61,66,76,78,82,85,93}
{0,2,24,29,41,44,45,50,57,64,75,103}
{0,2,27,28,46,58,74,79,90,96,99,103}
{0,2,28,32,35,44,49,50,69,73,100,103}
{0,2,28,47,59,63,72,73,83,90,105,110}
{0,2,30,39,51,52,55,59,64,70,87,97}
{0,2,32,36,39,40,52,57,58,67,81,100}
{0,2,37,38,41,46,57,59,66,72,89,99}
{0,2,40,43,53,57,75,76,82,87,102,103}
{0,2,40,47,55,62,79,82,83,88,99,101}
{0,3,4,5,12,18,31,34,70,74,81,91}
{0,3,4,5,13,24,31,54,68,76,82,99}
{0,3,4,5,23,51,64,72,78,80,89,104}
{0,3,4,7,21,33,67,74,76,82,87,92}
{0,3,4,7,26,31,36,42,44,51,85,97}
{0,3,4,9,16,23,34,62,70,72,94,99}
{0,3,4,9,20,22,32,34,72,79,87,94}
{0,3,4,10,20,39,44,58,81,88,90,103}
{0,3,4,11,13,30,54,66,76,77,82,97}
{0,3,4,16,21,22,31,45,64,75,77,107}
{0,3,4,16,25,53,55,69,79,96,102,107}
{0,3,4,16,32,39,41,43,49,58,63,93}
{0,3,4,17,39,48,58,82,88,90,95,100}
{0,3,4,39,41,53,63,80,86,93,95,106}
{0,3,5,9,20,41,59,71,72,87,94,106}
{0,3,5,16,20,43,51,72,80,86,104,105}
{0,3,5,16,41,53,62,69,70,92,96,102}
{0,3,5,19,22,26,32,37,50,61,62,70}
{0,3,7,8,14,16,35,40,55,58,89,101}
{0,3,7,9,19,24,49,62,63,83,85,103}
{0,3,7,9,19,28,50,51,65,85,90,103}
{0,3,7,11,23,36,41,42,44,51,68,100}
{0,3,7,12,18,35,45,59,61,89,98,110}
{0,3,7,13,18,31,42,43,51,92,95,97}
{0,3,7,14,29,39,41,47,81,90,94,95}
{0,3,7,15,17,42,43,61,73,89,94,105}
{0,3,7,33,35,43,46,73,77,96,97,102}
{0,3,7,37,39,50,69,83,92,93,98,110}
{0,3,8,14,26,39,41,48,58,90,91,95}
{0,3,8,19,21,28,34,51,61,73,75,110}
{0,3,8,20,27,42,43,55,73,94,105,109}
{0,3,9,10,28,34,42,63,71,94,98,109}
{0,3,9,11,13,28,29,44,51,56,65,90}
{0,3,9,11,24,25,29,44,46,56,63,86}
{0,3,9,20,25,41,53,71,72,97,99,107}
{0,3,9,20,34,35,42,47,61,63,65,71}
{0,3,10,25,34,36,42,50,63,91,109,110}
{0,3,10,48,54,56,62,71,85,89,90,101}
{0,3,11,13,39,43,46,55,60,61,80,84}
{0,3,11,19,25,34,47,48,52,54,64,90}
{0,3,11,24,26,33,56,70,75,94,104,110}
{0,3,11,24,29,49,63,64,86,95,105,107}
{0,3,11,29,31,51,52,65,90,95,105,107}
{0,3,12,17,18,37,41,68,71,79,81,107}
{0,3,12,18,22,44,45,52,61,73,98,109}
{0,3,13,17,35,36,42,47,62,63,71,73}
{0,3,13,24,25,29,43,52,58,60,66,104}
{0,3,13,24,43,44,49,50,58,60,72,76}
{0,3,13,25,56,59,74,79,98,100,106,107}
{0,3,13,39,44,46,55,61,69,73,74,93}
{0,3,14,19,24,26,32,56,66,75,97,110}
{0,3,14,46,63,70,72,73,78,91,103,107}
{0,3,15,20,42,44,52,80,91,98,105,110}
{0,3,15,32,38,46,60,83,90,101,109,110}
{0,3,16,22,29,30,31,34,54,64,71,75}
{0,3,17,19,22,63,71,72,83,96,101,107}
{0,3,17,29,63,70,72,78,83,88,107,110}
{0,3,17,32,37,38,48,60,84,101,103,110}
{0,3,18,23,42,44,50,51,55,58,68,80}
{0,3,19,20,24,33,67,73,75,85,100,107}
{0,3,19,23,24,56,66,73,75,88,100,106}
{0,3,20,27,35,42,80,82,92,94,105,110}
{0,3,21,40,41,45,53,59,68,70,75,101}
{0,3,21,51,56,65,71,73,75,82,98,110}
{0,3,22,27,32,38,40,47,81,93,107,110}
{0,3,23,33,40,44,80,83,96,102,109,110}
{0,3,24,49,58,63,70,85,86,101,103,105}
{0,3,24,50,60,62,66,67,80,89,95,103}
{0,3,28,51,58,68,70,85,89,90,103,105}
{0,3,30,34,53,54,59,68,71,75,101,103}
{0,3,34,46,56,59,63,64,70,72,91,96}
{0,3,38,42,54,56,64,65,70,71,90,101}
{0,3,39,43,50,60,80,83,84,85,92,98}
{0,3,41,43,51,52,67,72,78,79,97,101}
{0,3,43,49,51,53,67,72,79,80,94,105}
{0,3,44,52,53,64,77,82,88,92,95,109}
{0,4,5,7,23,32,38,46,58,68,94,105}
{0,4,5,11,13,32,37,52,55,86,98,108}
{0,4,5,16,26,29,36,74,80,82,88,97}
{0,4,5,18,20,26,29,54,77,84,94,96}
{0,4,5,18,27,33,41,49,52,73,99,109}
{0,4,5,21,24,28,35,50,60,62,68,102}
{0,4,5,24,42,45,55,81,86,88,97,103}
{0,4,5,37,47,54,56,69,81,87,92,95}
{0,4,6,9,14,26,33,48,49,61,79,100}
{0,4,6,16,21,46,59,60,80,82,100,108}
{0,4,6,16,25,47,48,62,82,87,100,108}
{0,4,6,16,42,63,66,74,82,88,97,110}
{0,4,7,8,20,25,26,35,49,68,79,81}
{0,4,7,8,20,29,57,59,73,83,100,106}
{0,4,7,13,24,29,45,57,75,76,101,103}
{0,4,7,15,28,33,53,67,68,90,99,109}
{0,4,7,15,33,35,55,56,69,94,99,109}
{0,4,7,16,21,22,41,45,72,75,83,85}
{0,4,7,17,29,60,63,78,83,102,104,110}
{0,4,7,18,50,67,74,76,77,82,95,107}
{0,4,7,21,23,26,67,75,76,87,100,105}
{0,4,7,23,24,28,37,71,77,79,89,104}
{0,4,8,11,22,54,71,78,80,81,86,99}
{0,4,8,20,33,38,39,41,48,65,97,108}
{0,4,9,15,32,42,56,58,86,95,107,108}
{0,4,10,15,28,39,40,48,89,92,94,108}
{0,4,10,19,22,24,35,60,72,81,88,89}
{0,4,10,21,47,57,69,77,83,92,108,110}
{0,4,11,21,41,44,45,46,53,59,72,75}
{0,4,11,26,36,38,44,78,87,91,92,108}
{0,4,12,14,39,40,58,70,86,91,102,108}
{0,4,12,18,27,29,34,60,70,73,91,110}
{0,4,13,14,24,31,46,51,52,54,80,99}
{0,4,13,23,30,46,51,52,54,66,86,97}
{0,4,13,47,53,55,65,80,87,91,94,110}
{0,4,14,17,55,57,65,66,81,86,92,93}
{0,4,15,17,20,26,27,45,51,59,80,88}
{0,4,15,36,54,66,67,82,89,101,106,109}
{0,4,16,18,26,27,32,33,52,63,73,76}
{0,4,16,29,34,35,37,44,61,93,104,107}
{0,4,16,35,61,63,64,69,84,91,101,102}
{0,4,18,27,33,35,41,79,86,89,99,110}
{0,4,18,29,49,61,63,64,69,85,92,102}
{0,4,19,21,31,38,61,86,89,95,97,110}
{0,4,20,23,28,34,46,59,61,68,78,110}
{0,4,22,23,29,34,49,50,58,60,98,101}
{0,4,23,24,29,38,41,45,71,73,81,84}
{0,4,26,27,34,43,55,80,91,93,96,105}
{0,4,27,35,56,64,70,88,89,95,98,100}
{0,4,30,32,40,43,70,74,93,94,99,108}
{0,4,31,34,42,44,70,74,77,86,91,92}
{0,4,34,36,47,66,80,89,90,95,107,108}
{0,4,39,42,52,63,82,83,88,89,97,99}
{0,4,40,43,56,62,69,70,71,74,94,104}
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{0,11,24,29,35,39,42,56,58,61,102,110}
{0,11,25,26,33,38,52,54,56,62,102,105}
{0,11,25,29,38,48,55,71,76,77,79,91}
{0,11,30,31,36,37,45,47,59,63,98,101}
{0,11,30,44,53,54,59,71,72,75,79,109}
{0,11,31,43,45,46,51,67,74,84,93,97}
{0,11,32,50,62,63,78,85,97,102,105,107}
{0,11,36,48,57,64,65,87,91,97,106,109}
{0,11,37,47,59,67,73,82,98,100,101,105}
{0,11,39,47,49,71,76,88,91,92,97,104}
{0,11,43,60,67,69,70,75,88,100,104,108}
{0,12,13,16,20,25,31,48,58,72,74,102}
{0,12,13,16,20,50,52,63,82,96,105,106}
{0,12,13,16,34,64,69,78,84,86,88,95}
{0,12,13,28,35,47,52,55,57,61,72,93}
{0,12,14,15,20,36,43,53,62,66,80,91}
{0,12,14,22,23,28,29,48,59,69,72,107}
{0,12,14,49,50,53,58,69,71,78,84,101}
{0,12,15,16,17,25,36,43,66,80,88,94}
{0,12,15,16,21,28,35,46,74,82,84,106}
{0,12,16,20,23,34,66,83,90,92,93,98}
{0,12,16,25,26,36,43,58,63,64,66,92}
{0,12,16,51,54,64,75,94,95,100,101,109}
{0,12,17,18,27,41,60,71,73,103,107,110}
{0,12,17,20,22,26,37,58,76,88,89,104}
{0,12,17,39,41,49,77,88,95,102,107,108}
{0,12,18,23,26,42,46,47,79,89,96,98}
{0,12,19,34,35,47,65,86,97,101,103,106}
{0,12,20,26,35,51,53,54,58,64,75,101}
{0,12,21,28,29,51,55,61,70,73,75,86}
{0,12,21,49,51,65,75,92,98,103,107,110}
{0,12,22,23,28,43,57,60,61,68,70,87}
{0,12,22,25,29,30,36,38,57,62,77,80}
{0,12,22,39,45,52,54,65,70,73,74,109}
{0,12,22,48,59,65,69,70,72,88,97,103}
{0,12,25,27,34,44,76,77,81,97,100,105}
{0,12,25,30,31,33,40,57,89,100,103,107}
{0,12,26,29,30,33,52,57,62,68,70,77}
{0,12,28,33,44,50,53,57,65,67,92,93}
{0,12,28,35,37,39,45,54,59,89,107,110}
{0,12,29,35,43,57,80,87,98,106,107,108}
{0,12,30,31,56,58,66,70,73,79,90,95}
{0,12,30,51,62,66,68,71,76,88,95,110}
{0,12,31,57,59,60,65,80,87,97,98,107}
{0,12,32,43,57,61,70,80,87,103,108,109}
{0,12,36,53,55,62,63,66,80,95,100,101}
{0,12,37,48,50,53,62,68,72,94,95,102}
{0,12,43,46,61,66,85,87,93,94,98,101}
{0,12,46,53,55,61,66,71,90,93,94,97}
{0,13,14,17,28,33,38,40,46,70,80,89}
{0,13,14,18,20,30,56,77,80,88,96,102}
{0,13,14,18,33,35,45,52,75,100,103,109}
{0,13,14,34,36,54,62,65,69,71,81,86}
{0,13,15,21,24,49,72,79,89,91,106,110}
{0,13,15,22,32,64,65,69,85,88,93,99}
{0,13,15,22,45,59,64,83,93,99,100,103}
{0,13,16,52,56,63,73,93,96,97,98,105}
{0,13,18,19,21,28,45,77,88,91,95,99}
{0,13,18,24,28,31,45,47,50,91,99,100}
{0,13,18,38,52,53,75,84,94,96,100,103}
{0,13,19,26,27,28,31,51,61,68,72,108}
{0,13,21,24,25,31,41,60,65,79,102,109}
{0,13,21,24,28,30,40,49,71,72,86,106}
{0,13,21,27,29,38,53,60,63,64,65,83}
{0,13,22,28,36,44,47,68,94,104,106,110}
{0,13,24,25,33,74,77,79,93,96,100,106}
{0,13,25,29,33,36,47,79,96,103,105,106}
{0,13,25,31,36,39,55,59,60,92,102,109}
{0,13,35,44,54,78,84,86,91,96,107,110}
{0,13,38,43,53,55,59,62,70,88,90,110}
{0,13,41,59,60,61,64,71,86,95,97,103}
{0,14,15,22,27,41,43,45,51,91,94,100}
{0,14,15,37,46,56,58,62,65,73,86,91}
{0,14,16,18,24,64,67,73,84,98,99,106}
{0,14,16,19,60,68,69,80,93,98,104,108}
{0,14,16,44,53,65,66,69,73,78,84,101}
{0,14,17,18,21,40,45,50,56,58,65,99}
{0,14,17,18,25,27,44,68,80,90,91,96}
{0,14,17,21,27,32,45,56,57,65,106,109}
{0,14,18,19,30,40,43,50,88,94,96,102}
{0,14,18,27,37,44,60,65,66,68,80,100}
{0,14,19,26,27,41,52,58,61,101,107,109}
{0,14,19,38,48,54,55,58,66,79,81,88}
{0,14,22,28,45,57,60,61,62,70,81,88}
{0,14,23,24,29,41,42,45,49,79,81,92}
{0,14,23,29,31,37,75,82,85,95,106,107}
{0,14,24,41,47,52,56,59,60,72,81,109}
{0,14,25,31,34,74,80,82,84,98,103,110}
{0,14,25,45,57,59,60,65,81,88,98,107}
{0,14,26,60,67,69,75,80,85,104,107,108}
{0,14,29,34,35,45,57,81,98,100,107,108}
{0,14,33,44,46,76,80,83,84,96,101,102}
{0,14,34,39,52,60,63,67,69,79,88,110}
{0,14,37,44,46,59,67,70,71,77,87,106}
{0,14,37,44,55,63,64,65,68,80,97,103}
{0,15,16,24,26,64,67,77,81,99,100,106}
{0,15,16,28,46,67,78,82,84,87,92,104}
{0,15,16,31,33,35,41,44,65,90,99,104}
{0,15,16,31,38,43,52,77,98,101,107,109}
{0,15,17,19,25,28,49,74,83,88,95,110}
{0,15,17,27,34,57,82,85,91,93,106,107}
{0,15,18,49,61,71,74,78,79,85,87,106}
{0,15,19,20,33,35,41,44,69,92,99,109}
{0,15,20,21,23,49,68,80,84,93,94,104}
{0,15,20,21,31,43,67,84,86,93,94,97}
{0,15,20,26,27,45,49,59,62,100,102,110}
{0,15,20,39,41,47,48,52,55,65,77,108}
{0,15,22,25,26,27,45,73,86,94,100,102}
{0,15,22,26,29,45,46,50,59,93,99,101}
{0,15,22,27,36,61,82,85,91,93,95,110}
{0,15,22,32,33,42,46,58,77,103,105,106}
{0,15,22,34,39,42,44,48,59,80,98,110}
{0,15,24,26,32,40,53,81,99,100,101,104}
{0,15,25,27,33,67,76,80,81,97,100,104}
{0,15,29,32,33,40,42,59,83,95,105,106}
{0,16,17,21,30,64,70,72,82,97,104,108}
{0,16,18,19,23,29,40,66,76,88,96,102}
{0,16,19,23,30,45,55,57,63,97,106,110}
{0,16,19,24,30,42,55,57,64,74,106,107}
{0,16,20,21,53,63,70,72,85,97,103,108}
{0,16,21,22,24,36,56,67,81,85,94,104}
{0,16,21,32,38,41,45,53,55,80,81,99}
{0,16,23,25,27,33,42,47,77,95,98,99}
{0,16,23,33,42,46,60,71,91,103,105,106}
{0,16,25,31,39,51,61,87,98,104,108,109}
{0,16,28,29,32,50,80,85,94,100,102,104}
{0,16,28,46,47,72,74,82,86,89,95,106}
{0,17,19,26,27,30,44,59,64,65,75,87}
{0,17,20,21,26,37,39,49,51,89,96,104}
{0,17,23,28,32,35,36,48,57,85,87,101}
{0,17,23,30,32,43,48,51,52,87,89,101}
{0,17,23,31,45,68,75,86,94,95,96,99}
{0,17,24,26,27,32,45,57,61,65,68,79}
{0,17,24,32,39,77,79,89,91,102,107,108}
{0,17,27,39,41,76,77,80,85,96,98,105}
{0,17,27,41,43,71,80,92,93,96,100,105}
{0,17,29,32,33,34,42,53,60,83,97,105}
{0,17,41,53,63,64,69,84,98,101,102,109}
{0,17,49,60,63,67,71,83,96,101,102,104}
{0,18,19,20,23,30,45,54,56,62,70,83}
{0,18,19,25,28,30,41,45,68,76,97,105}
{0,18,19,25,30,45,46,54,56,94,97,107}
{0,18,19,44,46,54,58,61,67,78,83,99}
{0,18,20,40,41,54,79,84,94,96,100,103}
{0,18,21,22,34,50,57,59,61,67,76,81}
{0,18,21,31,57,62,64,73,79,87,91,92}
{0,18,22,32,35,73,75,83,84,99,104,110}
{0,18,24,32,53,61,84,88,99,101,104,110}
{0,18,26,29,33,35,45,50,75,88,89,109}
{0,18,30,31,46,53,65,70,73,75,79,90}
{0,18,30,46,51,62,68,71,75,83,85,110}
{0,18,37,38,42,50,56,65,67,72,98,108}
{0,18,39,50,54,56,59,64,76,83,98,99}
{0,18,46,59,67,73,75,84,99,106,109,110}
{0,18,48,53,62,68,70,72,79,95,107,108}
{0,19,20,24,32,38,47,49,54,80,90,93}
{0,19,20,25,26,34,36,48,52,87,90,100}
{0,19,20,25,34,37,41,67,69,77,80,107}
{0,19,21,27,28,32,35,45,57,88,91,106}
{0,19,22,23,26,40,52,86,93,95,101,106}
{0,19,23,50,53,61,63,89,93,96,105,110}
{0,19,24,29,35,37,44,78,90,104,107,108}
{0,19,24,38,61,68,70,83,91,94,95,101}
{0,19,24,39,42,73,85,95,98,102,103,109}
{0,19,29,35,36,39,47,60,62,69,92,106}
{0,19,30,32,62,66,69,70,82,87,88,97}
{0,19,30,40,43,78,82,94,96,104,105,110}
{0,19,31,35,44,45,55,62,77,82,83,85}
{0,19,33,42,43,48,60,61,64,68,98,100}
{0,19,37,40,50,76,81,83,92,98,106,110}
{0,19,45,47,48,53,68,75,85,86,95,99}
{0,20,21,34,59,64,74,76,80,83,91,109}
{0,20,22,40,48,51,55,57,67,72,97,110}
{0,20,23,24,25,32,38,51,54,90,94,101}
{0,20,25,38,46,49,53,55,65,74,96,97}
{0,20,30,37,41,77,80,93,99,106,107,108}
{0,20,31,45,49,58,68,75,91,96,97,99}
{0,20,32,34,35,40,56,63,73,82,86,100}
{0,20,34,35,57,66,76,78,82,85,93,106}
{0,21,24,30,32,34,49,50,65,72,77,86}
{0,21,24,32,40,46,55,68,69,73,75,85}
{0,21,29,35,53,54,60,63,65,76,80,103}
{0,21,29,52,56,67,69,72,78,79,97,103}
{0,21,32,36,38,41,46,58,65,80,81,93}
{0,21,39,51,52,67,74,86,91,94,96,100}
{0,21,46,55,60,67,82,83,98,100,102,108}
{0,21,47,57,59,63,64,77,86,92,100,108}
{0,22,23,30,39,51,76,87,89,92,101,107}
{0,22,23,37,57,62,75,83,86,90,92,102}
{0,22,24,32,60,71,78,85,90,91,94,106}
{0,22,26,32,41,44,46,57,82,94,103,110}
{0,22,27,39,42,43,48,55,62,73,101,109}
{0,22,31,41,43,47,50,58,71,76,96,110}
{0,22,31,41,65,71,73,78,83,94,97,98}
{0,22,35,36,39,50,55,60,62,68,92,102}
{0,23,27,38,40,43,49,50,68,74,82,103}
{0,23,30,32,45,53,56,57,63,73,92,97}
{0,23,30,40,42,57,61,62,75,77,83,86}
{0,23,30,41,49,50,51,54,66,83,89,97}
{0,23,31,52,60,66,84,85,91,94,96,107}
{0,23,37,42,61,71,77,78,81,89,102,104}
{0,23,37,45,51,68,80,83,84,85,93,104}
{0,23,48,51,57,59,72,73,77,92,94,104}
{0,24,30,32,37,42,53,56,57,70,92,101}
{0,24,34,43,65,78,79,82,93,98,103,105}
{0,24,36,46,47,52,67,81,84,85,92,94}
{0,24,41,43,50,51,54,68,83,88,89,99}
{0,25,26,44,56,72,77,88,94,97,101,109}
{0,25,27,35,39,42,48,59,64,80,92,110}
{0,25,28,34,36,49,50,54,69,71,81,88}
{0,25,30,40,42,46,49,57,75,77,97,98}
{0,25,34,39,46,61,62,77,79,81,87,90}
{0,25,36,38,41,50,56,60,82,83,90,99}
{0,25,37,46,53,54,76,80,86,95,98,100}
{0,25,38,39,59,61,79,87,90,94,96,106}
{0,25,46,49,55,57,59,74,75,90,97,102}
{0,25,48,55,65,67,82,86,87,100,102,108}
{0,26,28,29,34,49,56,66,67,76,80,92}
{0,26,28,36,39,66,70,89,90,95,104,107}
{0,26,30,33,42,47,48,67,71,98,101,109}
{0,26,31,33,42,48,56,60,61,80,98,101}
{0,26,36,38,42,43,56,65,71,79,87,90}
{0,26,36,39,57,76,77,81,89,95,104,106}
{0,26,36,48,56,62,71,87,89,90,94,100}
{0,26,37,43,47,48,50,66,75,81,89,101}
{0,26,45,57,61,70,71,81,88,103,108,109}
{0,26,47,50,58,66,72,81,94,95,99,101}
{0,27,30,38,40,66,70,73,82,87,88,107}
{0,27,31,50,51,56,65,68,72,98,100,108}
{0,28,30,44,54,71,77,82,86,89,90,102}
{0,28,36,38,60,65,77,80,81,86,93,100}
{0,28,37,49,50,53,57,62,68,85,95,109}
{0,28,39,46,53,58,59,62,74,79,101,103}
{0,28,41,49,55,57,66,81,88,91,92,93}
{0,28,46,47,48,51,58,73,82,84,90,98}
{0,30,32,43,62,76,85,86,91,103,104,107}
{0,30,34,37,38,50,55,56,65,79,98,109}
{0,30,35,44,50,52,54,61,77,89,90,93}
{0,30,48,51,52,64,80,87,89,91,97,106}
{0,31,34,49,54,73,75,81,82,86,89,99}
{0,31,43,53,56,60,61,67,69,88,93,108}
{0,32,33,37,53,56,61,67,79,92,94,101}
{0,32,42,49,51,64,76,82,87,90,106,110}
{0,32,43,46,50,54,66,79,84,85,87,94}
{0,32,49,56,58,59,64,77,89,93,97,100}
{0,34,40,42,52,67,74,78,81,97,98,102}
{0,34,41,43,49,54,59,78,81,82,85,99}
{0,34,43,47,48,64,67,71,78,93,103,105}
{0,34,46,60,63,64,67,86,91,96,102,104}
{0,35,36,39,44,55,57,64,70,87,97,109}
{0,35,37,49,59,76,82,89,91,102,107,110}
{0,35,38,48,59,78,79,84,85,93,95,107}
{0,35,39,51,53,61,62,67,68,87,98,108}
{0,36,39,52,58,65,66,67,70,90,100,107}
{0,36,40,47,57,77,80,81,82,89,95,108}
{0,38,40,48,49,64,69,75,76,94,98,108}
{0,38,40,50,52,63,68,69,72,89,96,104}
{0,38,41,51,55,73,74,80,85,100,101,109}
{0,38,44,46,52,61,75,79,80,91,101,104}
{0,38,45,48,58,69,70,74,88,97,103,105}
{0,38,45,53,60,77,80,81,86,97,99,109}
{0,40,43,49,60,74,75,82,87,101,103,105}
{0,40,46,48,50,64,69,76,77,91,102,108}
{0,41,44,46,60,63,67,73,78,91,102,103}
{0,41,49,50,61,74,79,85,89,92,106,108}
 

[Ed Pegg]

Mistake in one of my programs.  Your list boils down to 36 partitions.  phi(111)=72, so 36 looks correct with reversals.

{{1,1,3,7,15,9,2,6,8,13,28,18}, {1,1,3,12,17,6,8,14,23,7,11,8}, {1,1,3,20,10,7,4,36,3,13,6,7}, {1,2,7,17,32,11,3,4,4,12,13,5}, {1,2,12,20,11,14,4,9,10,7,16,5}, {1,2,16,9,6,8,12,10,26,11,6,4}, {1,2,26,19,12,4,9,1,10,7,15,5}, {1,3,4,5,6,17,10,14,2,28,9,12}, {1,3,4,30,2,11,19,14,9,1,5,12}, {1,3,5,11,2,7,6,17,10,12,2,35}, {1,3,8,13,2,7,23,14,5,19,10,6}, {1,3,11,5,5,2,6,24,10,9,22,13}, {1,3,12,5,22,2,8,28,11,7,7,5}, {1,3,14,12,34,7,2,6,5,5,19,3}, {1,3,14,15,5,1,10,12,24,17,2,7}, {1,3,17,7,8,7,38,2,10,2,11,5}, {1,3,18,30,5,9,6,2,2,7,16,12}, {1,4,2,10,26,21,3,8,8,6,9,13}, {1,4,3,10,12,31,3,15,5,19,2,6}, {1,4,8,6,9,2,5,26,10,3,18,19}, {1,4,9,34,6,2,10,15,7,4,3,16}, {1,4,14,9,6,2,6,38,7,3,10,11}, {1,4,15,2,10,7,23,25,3,6,2,13}, {1,4,16,3,5,6,12,13,2,7,10,32}, {1,5,1,8,2,12,4,35,3,10,11,19}, {1,5,9,3,4,26,2,8,3,27,4,19}, {1,6,3,2,11,4,23,8,21,8,6,18}, {1,6,5,15,1,8,2,38,3,10,4,18}, {1,7,5,14,2,2,6,40,3,6,11,14}, {1,7,9,12,25,11,2,3,9,6,4,22}, {1,8,41,3,2,14,3,4,6,5,13,11}, {1,12,18,21,11,4,2,3,5,12,7,15}, {1,13,25,5,10,2,4,3,8,18,2,20}, {1,14,20,5,13,8,3,4,2,10,9,22}, {1,15,2,2,6,3,21,25,9,5,7,15}, {1,18,12,16,5,11,6,3,4,8,2,25}}

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[Christian Sievers]

Before you wrote this, I tried n=94 with 11 elements. n=106 wasn't successful for over 2h, but for n=109 I got this difference base:

? b=[0,1,24,25,34,36,38,41,56,64,85,91]; \\ this is using PARI
? #b \\ size
%2 = 12
? #setbinop((x,y)->(x-y)%109,b,b)  \\ size of the set of differences mod 109
%3 = 109

The solver I'm using is "clingo" which - for Debian - is in the "gringo" package.

This is the latest version of the script I'm using:
------------- dbp.cl ----------------
m(0).
m(1).
{m(2..(n-1))}=s-2.
c(N):-N=0..n-1,m(N1),m(N2),N=N1-N2.
c(N):-N=0..n-1,m(N1),m(N2),n-N=N1-N2.
:- N=0..(n-1), not c(N).
#show m/1.
-------------------------------------

It is called like this:
clingo -t 8 --configuration=crafty -c s=12 -c n=109 dbp.cl

For a small example, try this:

clingo -c s=6 -c n=28 dbp.cl

Or add "-n 0" to see all (two) solutions that contain 0 and 1.

-t <n> gives the number of threads used (the base for n=111 was found with -t 6).
(I have 6 cores pretending to be 12, but didn't want to throw all computing power to this.)
I tried several configurations, "crafty" was the first where the computation for a big case terminated before I gave up. A serious attempt should certainly do more experiments with medium sized problems to find a good configuration.
The two -c options set the parameter n and s (size).
(A modified script could search a smallest set instead of one with a given size.)

[Ed Pegg]

I'm fairly sure these are correct:  basis form

{101, 12, {0,1,10,18,22,39,41,44,50,57,77,87}},
{102, 12, {0,1,4,16,19,40,42,50,59,64,70,77}},
{103, 12, {0,1,3,17,29,31,50,56,61,65,83,96}},
{104, 12, {0,1,3,15,17,42,46,55,65,72,83,99}},
{105, 12, {0,1,3,12,20,34,38,53,60,66,76,81}},
{106, 12, {0,1,3,9,19,39,53,61,66,83,90,94}},
{107, 12, {0,1,9,19,24,31,52,56,58,69,72,98}},
{108, 12, {0,1,3,15,21,46,50,59,69,76,87,103}},
{109, 12, {0,1,10,12,14,17,32,40,61,67,85,86}},
{110, 12, {0,1,5,7,15,24,33,36,57,68,70,95}},
{111, 12, {0,1,3,15,35,46,60,64,73,83,90,106}}  

{113, 12, {1, 3, 3, 8, 10, 5, 23, 17, 9, 2, 20, 12}}  partition form.  
{1,6,5,38,9,10,3,14,2,2,15,8}, {1,10,11,21,16,2,26,3,4,5,8,6}

I believe 113 might be the last 12, but I'm not sure.  

Using greedy methods, it's easy to find 13's for 112 and 114-118, but I don't entirely trust them.