A question about the metal mean sequences.

[djoyce099]

The metal mean sequences.

Not comparing the first column but all the
rest of the columns is the (5) in the second column
the only repeating integer with the (5) in the golden
mean sequence (Fibonacci sequence)?

1,1,2,3,5,8,13,21,34,55,89,144...
2,5,12,29,70,169,408,985,2378...
3,10,33,109,360,1189,3927,12970...
4,17,72,305,1292,5473,23184,98209...
5,26,135,701,3640,18901,98145,509626...
6,37,228,1405,8658,53353,328776,2026009...
7,50,357,2549,18200,129949,927843,6624850...
8,65,528,4289,34840,283009,2298912,18674305...
9,82,747,6805,61992,564733,5144589,46866034,426938895...
10,101,1020,10301,104030,1050601,10610040,107151001,1082120050...
11,122,1353,15005,166408,1845493,20466831,226980634,2517253805...
12,145,1752,21169,255780,3090529,37342128,451196065,5451694908...
13,170,2223,29069,380120,4970629,64998297,849948490...
14,197,2772,39005,548842,7722793,108667944,1529074009...
15,226,3405,51301,772920,11645101,175449435,2643386626...
16,257,4128,66305,1065008,17106433,274767936,4413393409...
17,290,4947,84389,1439560,24556909,418907013,7145976130...
18,325,5868,105949,1912950,34539049,623615832,11259624025...
19,362,6897,131405,2503592,47699653,908796999,17314842634...
20,401,8040,161201,3232060,64802401,1299280080,26050404001...
...

I believe there is no other repeats except (5) no matter how far you extend
these sequences down or across.
Not comparing the first column of course.

Dan

[djoyce099]

It creates this chaotic sequence below where only the 5 is repeated
and the first column is also eliminated.

1,1,2,3,5,5,8,10,12,13,17,21,26,29,33,34,37,50,55,65,70,72,82,89,101,
109,122,135,144,145,169,170,197,226,228,233,257,290,305,325,357,360,362,
377,401,408,442...

[djoyce099]

Can anyone prove that any other number other than (5) repeats somewhere in this sequence?

[Barry Schwarz]

What do you think "it" is and how does it create anything?

>and the first column is also eliminated.
>
>1,1,2,3,5,5,8,10,12,13,17,21,26,29,33,34,37,50,55,65,70,72,82,89,101,
>109,122,135,144,145,169,170,197,226,228,233,257,290,305,325,357,360,362,
>377,401,408,442...

What you appear to have done is take all the numbers in your original
matrix, except for the first column, and sorted them. I guess this
simplifies the process of checking for duplicates. Not a proof but,
if you used the rules I describe below, then to the limit of 64 bit
unsigned integers "5" is the only duplicate.

But that begs a question. Why didn't you tell us how you generated
the values? Can you confirm that the following is correct:
Using normal (i,j) matrix notation:
For row 1, (1,j) is the j-th Fibonacci number
For other rows
(i, 1) = i
(i, 2) = (i, 1)^2+1 = i^2+1
(i, j) = (i, 1) * (i, j-1) + (i, j-2)

Which begs a second question. Why did you use a special rule for row
1? If you had used the same rule as the other rows, the sequence
would be 1, 2, 3, 5, 8, 13,... which is the Fibonacci sequence without
the duplicate "1". It doesn't change your speculation that 5 is the
only duplicate. It does remove a supposed mystical connection to
Fibonacci or the golden mean and the fact that the sqrt of 5 appears
in closed form expression for that value.

--
Remove del for email

[djoyce099]

Example. taken each integer (n) from column 1.

1/((sqrt(n^2 +4) - n)/2) to create the metal mean.

Starting with the silver mean where n=2
1/((sqrt(2^2 +4) -2)/2) = 2.4142135623... = the silver mean.

2.4142135623...*2 = 4.828... round up too 5
2.4142135623...*5 = 12.0710... round down too 12
2.4142135623...*12 = 28.9705... round up too 29
2.4142135623...*29 = 70.01219... round down too 70
2.4142135623...*70 = 168.9949... round up too 169
... each future term converges closer to that term.
As with the Fibonacci sequence and the golden means ---1.618033...
So the silver mean sequence = 2,5,12,29,70...

And so forth where n =3 and the bronze means = 3.3027756377...
Then --- 3.3027756377...*3 = 9.90832... round up to 10
...
Thus the bronze mean sequence = 3,10,33,109,...

> Which begs a second question. Why did you use a special rule for row
> 1? If you had used the same rule as the other rows, the sequence
> would be 1, 2, 3, 5, 8, 13,... which is the Fibonacci sequence without
> the duplicate "1". It doesn't change your speculation that 5 is the
> only duplicate. It does remove a supposed mystical connection to
> Fibonacci or the golden mean and the fact that the sqrt of 5 appears
> in closed form expression for that value.

Just to avoid all possible duplications from column 1.

[Barry Schwarz]

On Wed, 13 Jan 2021 14:16:10 -0800 (PST), djoyce099
<hlauk.h...@gmail.com> wrote:

>Example. taken each integer (n) from column 1.
>
>1/((sqrt(n^2 +4) - n)/2) to create the metal mean.

Where did this definition of metal mean come from? Where did the term
"metal mean" come from?

>Starting with the silver mean where n=2

Where did the term "silver mean" come from?

[djoyce099]

Sorry.
Should have stated, Metalic means. Wikipedia ---

Metallic means (Metallic ratios) Class
N Ratio Value (Type)
0: 0 + √4/2 1
1: 1 + √5/2 1.618033989[a] Golden
2: 2 + √8/2 2.414213562[b] Silver
3: 3 + √13/2 3.302775638[c] Bronze
4: 4 + √20/2 4.236067978[d]
5: 5 + √29/2 5.192582404[e]
6: 6 + √40/2 6.162277660[f]
7: 7 + √53/2 7.140054945[g]
8: 8 + √68/2 8.123105626[h]
9: 9 + √85/2 9.109772229[i]
⋮
n: n + √4 + n2/2

[djoyce099]

correction -- Metallic. with two LL's

[djoyce099]

Where (n) = an integer >5 then there will never be
a Fibonacci number of the form --- n^2+1

[djoyce099]

Correction --
Where (n) = an integer >2 then there will never be

[Peter]

I know it's none of my business, but may I make a suggestion? When you
compose a post, but before you post it, pretend that you are someone
else. Read your post and ask "does this make sense?" Note that that's
not "does this make sense to Dan Joyce?", but does it make sense to
someone who is not privy to the contents of your mind?

--
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays

[djoyce099]

Just refute what I post with proof that it isn't true even though I have no proof of the statement
but just a conjecture.
There are many such examples in math where there is no proof either way but just a conjecture.
So just what is wrong with my post as being a conjecture?

[James Waldby]

djoyce099 <hlauk.h...@gmail.com> wrote:
> The metal mean sequences.
>
> Not comparing the first column but all the
> rest of the columns is the (5) in the second column
> the only repeating integer with the (5) in the golden
> mean sequence (Fibonacci sequence)?
>
> 1,1,2,3,5,8,13,21,34,55,89,144...
> 2,5,12,29,70,169,408,985,2378...
> 3,10,33,109,360,1189,3927,12970...
> 4,17,72,305,1292,5473,23184,98209...
> 5,26,135,701,3640,18901,98145,509626...

(snip 15 lines of other sequence examples)

> I believe there is no other repeats except (5) no matter how
> far you extend these sequences down or across.
> Not comparing the first column of course.

Let M_k denote the kth sequence, with elements e_0, e_1, ..., in which
e_0 and e_1 are 0 and 1, and thereafter e_(j+1) = k*e_j + e_(j-1). (In
the terminology of <http://www.mi.sanu.ac.rs/vismath/spinadel/>, these
are "generalized secondary Fibonacci sequences" with sequence
parameters p and q -- the multipliers of e_j and e_(j-1) -- equal to k
and 1. (The article <https://en.wikipedia.org/wiki/Metallic_mean>
mentions <https://oeis.org/A006190> where it's noted that the
sequences give denominators of continued fractions converging to the
related metal means, but otherwise I think the wikipedia article is
less clear than Spinadel.)

Anyhow, I don't see an obvious reason why there should be "no other
repeats except (5)". Perhaps someone can mention an obvious reason?

My computer testing apparently agrees with yours about there being no
other duplicates among small values from the sequences. I ran a test
for 0 < k < 1000000 and my program found no duplicates in the first
billion numbers of the merged sequences. The last numbers among this
first billion with k less than a million are near 10^5531.62 in size.

[James Waldby]

djoyce099 <hlauk.h...@gmail.com> wrote:
> On Monday, January 18, 2021 at 12:44:40 PM UTC-5, Peter wrote:
>> djoyce099 wrote:
>> > The metal mean sequences.
>> >
>> > Not comparing the first column but all the
>> > rest of the columns is the (5) in the second column
>> > the only repeating integer with the (5) in the golden
>> > mean sequence (Fibonacci sequence)?
>> >
>> > 1,1,2,3,5,8,13,21,34,55,89,144...
>> > 2,5,12,29,70,169,408,985,2378...
>> > 3,10,33,109,360,1189,3927,12970...
>> > 4,17,72,305,1292,5473,23184,98209...
>> > 5,26,135,701,3640,18901,98145,509626...

(snip 15 more)

>> >
>> > I believe there is no other repeats except (5) no matter
>> > how far you extend these sequences down or across.
>> > Not comparing the first column of course.

...

>> I know it's none of my business, but may I make a suggestion? When you
>> compose a post, but before you post it, pretend that you are someone
>> else. Read your post and ask "does this make sense?" Note that that's
>> not "does this make sense to Dan Joyce?", but does it make sense to
>> someone who is not privy to the contents of your mind?
>
> Just refute what I post with proof that it isn't true even though
> I have no proof of the statement but just a conjecture.
> There are many such examples in math where there is no proof either
> way but just a conjecture.
> So just what is wrong with my post as being a conjecture?

The problem with it is lack of context and lack of explanation. For
example, instead of "The metal mean sequences." you could write,
"Beginnings of the first 20 metal mean sequences are shown below. See
definitions in..." and then provide a couple of helpful references.

The question about whether 5 is "the only repeating integer" while
"Not comparing the first column" is clear enough but perhaps you could
explain why you think so. It appears to be an interesting question
and worth posting about, using understandable sentences.

--
jiw

[Peter]

How does one refute what one does not understand?

> what I post with proof that it isn't true even though I have no proof of the statement
> but just a conjecture.
> There are many such examples in math where there is no proof either way but just a conjecture.
> So just what is wrong with my post as being a conjecture?
>
>> When, once, reference was made to a statesman almost universally
>> recognized as one of the villains of this century, in order to
>> induce him to a negative judgment, he replied: "My situation is
>> so different from his, that it is not for me to pass judgment".
>> Ernst Specker on Paul Bernays

--

[djoyce099]

Thanks for your input.
Interesting that size a number is even possible in your search.
This will remain an open question like the Fermat primes and many other conjectures.

[djoyce099]

I guess I assume too much about people understanding my post but try
to shed light on my reasoning with this response to Barry about why I
left out column (1) ----

"Just to avoid all possible duplications from column 1."

Dan

[Sergio]

On 1/12/2021 2:20 PM, djoyce099 wrote:
> The metal mean sequences.
>
> Not comparing the first column but all the
> rest of the columns is the (5) in the second column
> the only repeating integer with the (5) in the golden
> mean sequence (Fibonacci sequence)?
>
> 1,1,2,3,5,8,13,21,34,55,89,144...

that is fibonacci without the 0

> 2,5,12,29,70,169,408,985,2378...

that is a Pell number without the 0, 1,

> 3,10,33,109,360,1189,3927,12970...

Copper sequence without the 1

https://math.stackexchange.com/questions/3262735/fibonacci-sequence-and-other-metallic-sequences-emerged-in-the-form-of-fractions

[djoyce099]

James,
So does that mean the last number in this matrix that comparing all other
cells in this matrix would have 10^5531.62 digits?

[Bud Orchard]

Sergio wrote:

> On 1/12/2021 2:20 PM, djoyce099 wrote:
>> The metal mean sequences.
>> Not comparing the first column but all the rest of the columns is the
>> (5) in the second column the only repeating integer with the (5) in the
>> golden mean sequence (Fibonacci sequence)?
>> 1,1,2,3,5,8,13,21,34,55,89,144...
>
> that is fibonacci without the 0

be gone from white house, demon.

[djoyce099]

On Tuesday, January 12, 2021 at 3:20:52 PM UTC-5, djoyce099 wrote:

It creates this chaotic sequence below where only the 5 is repeated

and the first column is also eliminated.

It lists all integers < 1,000 for a total of 61 terms and rarer then
the primes.
As this number increases the rarety also increases dramatically.

2,3,5,5,8,10,12,13,17,21,26,29,33,34,37,50,55,65,70,72,82,89,101,
109,122,135,144,145,169,170,197,226,228,233,257,290,305,325,357,360,362,
377,401,408,442,485,528,530,577,610,626,677,701,730,747,785,842,901,962,
985,987

All integers < 2,000 = 81 terms total.

How many terms for all integers <10,000?

[Barry Schwarz]

Since the average distance between primes is just 15, that is not much
of a surprise.

>As this number increases the rarety also increases dramatically.
>
>2,3,5,5,8,10,12,13,17,21,26,29,33,34,37,50,55,65,70,72,82,89,101,
>109,122,135,144,145,169,170,197,226,228,233,257,290,305,325,357,360,362,
>377,401,408,442,485,528,530,577,610,626,677,701,730,747,785,842,901,962,
>985,987

If my algorithm generates your sequences correctly, there are only 60
terms < 1,000. 962 is bogus

>All integers < 2,000 = 81 terms total.

There are only 80 terms < 2,000

2, 3, 5, 5, 8, 10, 12,
13,
17, 21, 26, 29, 33, 34, 37,
50,
55, 65, 70, 72, 82, 89, 101,
109,
122, 135, 144, 145, 169, 170, 197,
226,
228, 233, 257, 290, 305, 325, 357,
360,
362, 377, 401, 408, 442, 485, 528,
530,
577, 610, 626, 677, 701, 730, 747,
785,
842, 901, 962, 985, 987, 1020, 1025,
1090,
1157, 1189, 1226, 1292, 1297, 1353, 1370,
1405,
1445, 1522, 1597, 1601, 1682, 1752, 1765,
1850,

>How many terms for all integers <10,000?

There are 157 terms < 10,000

2, 3, 5, 5, 8, 10, 12,
13,
17, 21, 26, 29, 33, 34, 37,
50,
55, 65, 70, 72, 82, 89, 101,
109,
122, 135, 144, 145, 169, 170, 197,
226,
228, 233, 257, 290, 305, 325, 357,
360,
362, 377, 401, 408, 442, 485, 528,
530,
577, 610, 626, 677, 701, 730, 747,
785,
842, 901, 962, 985, 987, 1020, 1025,
1090,
1157, 1189, 1226, 1292, 1297, 1353, 1370,
1405,
1445, 1522, 1597, 1601, 1682, 1752, 1765,
1850,
1937, 2026, 2117, 2210, 2223, 2305, 2378,
2402,
2501, 2549, 2584, 2602, 2705, 2772, 2810,
2917,
3026, 3137, 3250, 3365, 3405, 3482, 3601,
3640,
3722, 3845, 3927, 3970, 4097, 4128, 4181,
4226,
4289, 4357, 4490, 4625, 4762, 4901, 4947,
5042,
5185, 5330, 5473, 5477, 5626, 5741, 5777,
5868,
5930, 6085, 6242, 6401, 6562, 6725, 6765,
6805,
6890, 6897, 7057, 7226, 7397, 7570, 7745,
7922,
8040, 8101, 8282, 8465, 8650, 8658, 8837,
9026,
9217, 9303, 9410, 9605, 9802,

[djoyce099]

Why is it bogus? --- 31^2 +1 = 962 the first number in this second column.
All n in the first column from 1 to n creates the second column as n^2+1

> >All integers < 2,000 = 81 terms total.
> There are only 80 terms < 2,000
>
> 2, 3, 5, 5, 8, 10, 12,
> 13,
> 17, 21, 26, 29, 33, 34, 37,
> 50,
> 55, 65, 70, 72, 82, 89, 101,
> 109,
> 122, 135, 144, 145, 169, 170, 197,
> 226,
> 228, 233, 257, 290, 305, 325, 357,
> 360,
> 362, 377, 401, 408, 442, 485, 528,
> 530,
> 577, 610, 626, 677, 701, 730, 747,
> 785,
> 842, 901, 962, 985, 987, 1020, 1025,
> 1090,
> 1157, 1189, 1226, 1292, 1297, 1353, 1370,
> 1405,
> 1445, 1522, 1597, 1601, 1682, 1752, 1765,
> 1850,
> >How many terms for all integers <10,000?
> There are 157 terms < 10,000

What about 1937 where 44^2 +1 =1937
You also show 962 in this count of numbers <2000
??

1937 is also included in this count of all numbers <10,000

This is probably the most chotic sequence ever constructed
from an ordered sequence of related sequences. Being of course
all metallic mean sequences --->oo.

Thanks, Barry, for your input.

Dan

[Barry Schwarz]

On Sun, 24 Jan 2021 13:18:45 -0800 (PST), djoyce099

<hlauk.h...@gmail.com> wrote:

>On Sunday, January 24, 2021 at 1:32:32 AM UTC-5, Barry Schwarz wrote:
>> On Sat, 23 Jan 2021 14:06:52 -0800 (PST), djoyce099
>> <hlauk.h...@gmail.com> wrote:

<snip>

>> >It lists all integers < 1,000 for a total of 61 terms and rarer then
>> >the primes.
>> Since the average distance between primes is just 15, that is not much
>> of a surprise.
>> >As this number increases the rarety also increases dramatically.
>> >
>> >2,3,5,5,8,10,12,13,17,21,26,29,33,34,37,50,55,65,70,72,82,89,101,
>> >109,122,135,144,145,169,170,197,226,228,233,257,290,305,325,357,360,362,
>> >377,401,408,442,485,528,530,577,610,626,677,701,730,747,785,842,901,962,
>> >985,987
>> If my algorithm generates your sequences correctly, there are only 60
>> terms < 1,000. 962 is bogus
>
>Why is it bogus? --- 31^2 +1 = 962 the first number in this second column.
>All n in the first column from 1 to n creates the second column as n^2+1

You are correct. My algorithm created the correct values. My report
routine failed to deal with the values in the last row.

>> >All integers < 2,000 = 81 terms total.
>> There are only 80 terms < 2,000
>>
>> 2, 3, 5, 5, 8, 10, 12,
>> 13,
>> 17, 21, 26, 29, 33, 34, 37,
>> 50,
>> 55, 65, 70, 72, 82, 89, 101,
>> 109,
>> 122, 135, 144, 145, 169, 170, 197,
>> 226,
>> 228, 233, 257, 290, 305, 325, 357,
>> 360,
>> 362, 377, 401, 408, 442, 485, 528,
>> 530,
>> 577, 610, 626, 677, 701, 730, 747,
>> 785,
>> 842, 901, 962, 985, 987, 1020, 1025,
>> 1090,
>> 1157, 1189, 1226, 1292, 1297, 1353, 1370,
>> 1405,
>> 1445, 1522, 1597, 1601, 1682, 1752, 1765,
>> 1850,

> What about 1937 where 44^2 +1 =1937
>You also show 962 in this count of numbers <2000

You are correct on both counts.

>??

>>> >How many terms for all integers <10,000?
>> There are 157 terms < 10,000

Because 10000 is a perfect square, the error in my report routine did
not invalidate the result. It is still 157

[djoyce099]

Thanks, Barry.

Have you any speculation on why, when eliminating column (1), there is no
repeat of any integers except the (5) in this sequence?
Not proven of course, but a possibility that it is true.
Could there be a scatter chart constructed that would explain the anomaly or some
other mathematical procedure that would explain it and therefore prove it?

[djoyce099]

The average gaps between terms gradually increase as each new term is added.
Like the primes but with odd and even-numbered gaps, unlike the primes.
Will, for instance, the last gap (1) appear between 169 and 170 and the last gap (2)
appear between 985 and 987?

[Barry Schwarz]

On Mon, 25 Jan 2021 18:16:45 -0800 (PST), djoyce099

The gap between primes does not appear to trend upward the further out
you go. For example, the largest gap in the first million primes is
154 between 4652353 and 4652507. (This gap also occurs twice more.)
Yet several thousand primes later, the gap between 15485609 and
15485611 is only 2.

>Will, for instance, the last gap (1) appear between 169 and 170 and the last gap (2)
>appear between 985 and 987?

[djoyce099]

Agree, that gap can be erratic with the primes but the average gap still increases as these
primes get larger.
Not so sure about this sequence. The gaps could be not as erratic as the primes as the average gaps
grow larger as the numbers get larger.

[James Waldby]

On Tue, 26 Jan 2021 06:04:15 -0800 (PST), djoyce099 wrote:
> On Tuesday, January 26, 2021 at 4:03:15 AM UTC-5, Barry Schwarz wrote:

>> On Mon, 25 Jan 2021 18:16:45 -0800 (PST), djoyce099 wrote:
>> >On Monday, January 25, 2021 at 1:35:31 PM UTC-5, djoyce099 wrote:
>> >> On Monday, January 25, 2021 at 4:41:28 AM UTC-5, Barry Schwarz wrote:

>> >> > On Sun, 24 Jan 2021 13:18:45 -0800 (PST), djoyce099 wrote:
>> >> > >On Sunday, January 24, 2021 at 1:32:32 AM UTC-5, Barry Schwarz wrote:

>> >> > >> On Sat, 23 Jan 2021 14:06:52 -0800 (PST), djoyce099 wrote:
[snip]

>> >> Have you any speculation on why, when eliminating
>> >> column (1), there is no repeat of any integers except
>> >> the (5) in this sequence? Not proven of course, but a
>> >> possibility that it is true. Could there be a scatter
>> >> chart constructed that would explain the anomaly or some
>> >> other mathematical procedure that would explain it and
>> >> therefore prove it?

There probably is a mathematical procedure to explain the
observed phenomena and to prove things about them. I don't know
what that procedure might be. Note, regarding whether any
integers repeat, see my post from Tue, 19 Jan 2021 06:29:51 UTC
in which I mentioned that for 0 < k < 1000000, my program found
no duplicates (other than 5) in the first billion numbers.

>> >The average gaps between terms gradually increase as each new term is added.
>> >Like the primes but with odd and even-numbered gaps, unlike the primes.
>> The gap between primes does not appear to trend upward the further out
>> you go. For example, the largest gap in the first million primes is
>> 154 between 4652353 and 4652507. (This gap also occurs twice more.)
>> Yet several thousand primes later, the gap between 15485609 and
>> 15485611 is only 2.
>> >Will, for instance, the last gap (1) appear between 169 and 170 and the last gap (2)
>> >appear between 985 and 987?

> Agree, that gap can be erratic with the primes but the average
> gap still increases as these primes get larger.

It is true the average gap increases, and it's no surprise. One
can easily prove it increases sharply.

> Not so sure about this sequence. The gaps could be not as
> erratic as the primes as the average gaps grow larger as the
> numbers get larger.

Your conjecture that "The gaps could be not as erratic as the primes",
seems plausible, albeit not necessarily well-defined. Anyhow,
regarding "the average gaps grow larger as the numbers get larger",
that is true of the sequence of primes as well as the merged sequence
of metal-mean sequences, so it looks to me like that fact by itself
can't predict whether any given gap can occur again. It probably is
possible to put bounds on the probability of finding some given gap
within any given section of the sequence.

Note, I ran a Python program for a couple of hours to go through the
first billion numbers in the merged sequences of M_1 ... M_8000000,
looking at gap sizes from 0 up to 10000. In this number set, only
about 5500 of the possible gap sizes occurred. For example, no gaps
of sizes 11, 19, 22, 28, 30, 41, 42, 44, 54, 55, 60 ... were found.

A few gap sizes occurred numerous times. Here are the four gaps that
occurred the most times in the whole sequence: 2: 9 times, in first 60
numbers; 5: 8 times, in first 41868 numbers; 257: 7 times, in first
1060542 numbers; 967: 7 times, in first 3192557 numbers. Also, my
program didn't find any gaps smaller than 10000 when looking through
the last 991 million of the first billion numbers.

--
jiw

[djoyce099]

On Wednesday, January 27, 2021 at 1:29:54 AM UTC-5, James Waldby wrote:
> On Tue, 26 Jan 2021 06:04:15 -0800 (PST), djoyce099 wrote:
> > On Tuesday, January 26, 2021 at 4:03:15 AM UTC-5, Barry Schwarz wrote:
> >> On Mon, 25 Jan 2021 18:16:45 -0800 (PST), djoyce099 wrote:
> >> >On Monday, January 25, 2021 at 1:35:31 PM UTC-5, djoyce099 wrote:
> >> >> On Monday, January 25, 2021 at 4:41:28 AM UTC-5, Barry Schwarz wrote:
> >> >> > On Sun, 24 Jan 2021 13:18:45 -0800 (PST), djoyce099 wrote:
> >> >> > >On Sunday, January 24, 2021 at 1:32:32 AM UTC-5, Barry Schwarz wrote:
> >> >> > >> On Sat, 23 Jan 2021 14:06:52 -0800 (PST), djoyce099 wrote:
> [snip]
> >> >> Have you any speculation on why, when eliminating
> >> >> column (1), there is no repeat of any integers except
> >> >> the (5) in this sequence? Not proven of course, but a
> >> >> possibility that it is true. Could there be a scatter
> >> >> chart constructed that would explain the anomaly or some
> >> >> other mathematical procedure that would explain it and
> >> >> therefore prove it?
> There probably is a mathematical procedure to explain the
> observed phenomena and to prove things about them. I don't know
> what that procedure might be. Note, regarding whether any
> integers repeat, see my post from Tue, 19 Jan 2021 06:29:51 UTC
> in which I mentioned that for 0 < k < 1000000, my program found
> no duplicates (other than 5) in the first billion numbers.

Yes, I read that, and thanks.

James,

Your last statement verifies my argument that these merged sequences gaps
that is not as erratic as the primes but their average gap increases as these
numbers and sequences continue to grow in size.

How many twin primes are there in the last 991 million of the first billion numbers
of the primes that can show how erratic the prime gaps really are compared to
this sequence's gaps? That is just looking at the twins so can you imagine the other small
single and double-digit gaps involved in that last 991 million of the first billion numbers
of the primes?

It also does give more weight to the fact there is no duplicate number other
then the 5.
Until a mathematical procedure is developed to prove or disprove this hypothesis
we all will remain in the dark.
This will be an interesting equation, if possible, for sure

I am not sure if the word hypothesis is the correct nomenclature here?

Dan

[djoyce099]

Maybe set up a Ulam type chart to possibly observe a pattern?

[djoyce099]

This, on a small observation of this special dannyj spiral, does prove there can never be any repeats other than the (5),

0 0 5 <---- This cell holds 2 5's
8 0 0
0 2 3
10 - -

Extended much further it shows a row of offset diagonal numbers from lower left to the upper right with a few outliers
to the left and right of this diagonal line. As this diagonal line extends much further these outliers become less
and less numerous where the integers are more and more concentrated on this offset diagonal line,

The first few outliers not on this diagonal line are for the first 52 numbers in this sequence ending in 677 which is
to the upper right of this offset diagonal line --
3,8,12,13,21,29,33,34,55,70,72,89,109,135,144,169,228,233,305,357,360,377,408,528,610.. The rest reside on the
offset diagonal line.

A good pixel challenge for Chris Thompson.
Just turn on those cells where that numbered cell is in the sequence. Just like in the prime number Ulam spiral.

Dan

[Peter]

What's one of those?

> , does prove there can never be any repeats other than the (5),
>
> 0 0 5 <---- This cell holds 2 5's
> 8 0 0
> 0 2 3
> 10 - -
>
> Extended much further it shows a row of offset diagonal numbers from lower left to the upper right with a few outliers
> to the left and right of this diagonal line. As this diagonal line extends much further these outliers become less
> and less numerous where the integers are more and more concentrated on this offset diagonal line,
>
> The first few outliers not on this diagonal line are for the first 52 numbers in this sequence ending in 677 which is
> to the upper right of this offset diagonal line --
> 3,8,12,13,21,29,33,34,55,70,72,89,109,135,144,169,228,233,305,357,360,377,408,528,610.. The rest reside on the
> offset diagonal line.
>
> A good pixel challenge for Chris Thompson.
> Just turn on those cells where that numbered cell is in the sequence. Just like in the prime number Ulam spiral.
>
> Dan
>