sums of distinct primes

[David Corneth]

Hi,

So I came across Paolo's new sequence A387198 called "Smallest integer
that can be expressed as the sum of k different primes, for all k’s
between 2 and n, with n >= 2." as of now (not yet approved, I think it
should be accepted).
I added some terms and it made me wonder.
Does the sequence:
"a(n) is the smallest k such that for any m >= k we have m is the sum
of exactly n distinct primes."
Having offset 3.
I think terms would be conjectured like the Goldbach conjecture.
First few terms I get are 18, 31, 42, 61.
So that would be:
Every integer >= 18 is the sum of exactly 3 distinct primes. etc.
But as of now it is conjected.
Having these terms would ease the computation of terms for A387198 and
the related A090700 (I extended the latter without using the
conjecture).

I also found A344989 where I'd say the 0's are conjectured.
Does it make sense to add:
"a(n) is the smallest k such that for any m >= k we have m is the sum
of exactly n distinct primes."
to OEIS?

Best,
David

[M F Hasler]

On Fri, Aug 22, 2025 at 8:52 AM David Corneth <davida...@gmail.com> wrote:

Does the sequence:
"a(n) is the smallest k such that for any m >= k we have m is the sum
of exactly n distinct primes."

I think that would make sense.

Together with the sequence that shows in how many ways a given m >= k

is the sum of exactly n primes (expected to grow steadily, similar to https://oeis.org/A002375/graph),

one would have a convincing argument that the conjectured values are correct with sufficiently high probability.

From the growth properties of these counting sequences (esp. the "lower bound" corresponding to "late birds"),

one should be able to get good ideas about how one can limit the search space generously enough

so that it should be "beyond any reasonable doubt" that no counter-example can be found.

That would allow to define the sequences rigorously,

i.e., "conjectured" could be replaced and/or made more precise by giving the chosen formula for the search limit for each a(n).

- Maximilian

 

Having offset 3.
I think terms would be conjectured like the Goldbach conjecture.
First few terms I get are 18, 31, 42, 61.
So that would be:
Every integer >= 18 is the sum of exactly 3 distinct primes. etc.
But as of now it is conjected.
Having these terms would ease the computation of terms for A387198 and
the related A090700 (I extended the latter without using the
conjecture).

I also found A344989 where I'd say the 0's are conjectured.
Does it make sense to add:
"a(n) is the smallest k such that for any m >= k we have m is the sum
of exactly n distinct primes."
to OEIS?

Best,
David

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[Neil Sloane]

I approved A387198.

You said "Does it make sense to add:

"a(n) is the smallest k such that for any m >= k we have m is the sum
of exactly n distinct primes." "

Me: Certainly!  

This is a rare case where a couple of commas would help!  

How about this definition: (2 commas, and "a" rather than "the")

"a(n) is the smallest k such that, for any m >= k,  m is a sum of exactly n distinct primes."

Best regards

Neil 

Neil J. A. Sloane, Chairman, OEIS Foundation.

Also Visiting Scientist, Math. Dept., Rutgers University, 

Email: njas...@gmail.com

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[David Corneth]

Okay thanks both! 

Yes I added the sequence using that name Neil. 

It is A387215. I managed to find some more terms but I'm not sure what's a good bound. 

For now I used a(n) = k where k is the smallest positive integer such that for any m, k <= m <= k + 3*n can be written as the sum of n distinct primes. 
But the k + 3*n is quite arbitrary. Happy to hear more thoughts on this. 

Maximilian, would the sequence you describe be a family of sequences? Like "a(n) is number of ways to write n as the sum of k primes."?

And then a sequence for k = 3, 4, 5,... or put them all in one sequence like "T(n, k) is the number of ways to write n as the sum of k distinct primes." with maybe n >= sum of first k primes.?

Best,

David

[M F Hasler]

On Fri, Aug 22, 2025, 20:23 David Corneth <davida...@gmail.com> wrote:

Maximilian, would the sequence you describe be a family of sequences? Like "a(n) is number of ways to write n as the sum of k primes."?

And then a sequence for k = 3, 4, 5,... or put them all in one sequence like "T(n, k) is the number of ways to write n as the sum of k distinct primes." with maybe n >= sum of first k primes.?

Sorry for the late reply... (I was and still am traveling...)

I didn't mean to (necessarily) submit those (other auxiliary) sequences, but rather to look at them to get an idea what a sufficiently "safe" search limit could be for individual a(n) and then maybe find a general formula for such a limit as function of n.

But yes, it would be a separate sequence for each n :

For given n, we want that:

for any m >= k = a(n) ,  m is the sum

of exactly n distinct primes." 

To be confident that we've searched far enough to exclude that there is a larger m which is not the sum of exactly m primes,

I'd consider the sequence b(n,m) = number of times m is the sum of exactly n primes.

Remember that we are looking for the last zero in each row n, which would be in column k–1, where k = a(n).

Computing a given row far enough being that last zero should give enough confidence that there will be no further zero. 

(Finally, yes, why not submit this table b(n,k) as a table/sequence on its own.

Note that the b(n,k) as I would suggest it has a different order of arguments compared to the T(.,.) you mentioned.)

- Maximilian

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