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A202650 Number of ways to write n = p + p(k) + p(m) with 0 < k <= m, where p is a prime and p(.) is the partition function (A000041). 2
0, 0, 0, 1, 2, 3, 4, 4, 6, 5, 7, 5, 7, 5, 10, 6, 10, 5, 12, 7, 13, 5, 13, 6, 15, 6, 15, 6, 15, 6, 13, 7, 15, 8, 17, 10, 14, 8, 14, 11, 12, 9, 13, 11, 14, 14, 16, 13, 16, 14, 15, 12, 12, 14, 16, 14, 13, 10, 14, 16, 15, 14, 18, 17, 15, 17, 14, 14, 15, 16, 14, 13, 15, 19, 18, 18, 16, 15, 13, 17, 18, 14, 19, 17, 19, 18, 18, 15, 21, 17, 22, 13, 17, 14, 20, 15, 19, 13, 15, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 3.

(ii) For any integer n > 2, |n - p(k)| is prime for some k = 1,...,n. Also, for any positive integer n not equal to 7, n + p(k) is prime for some k = 1,...,n.

We have verified part (i) of the conjecture for all n = 4, 5, ..., 2*10^7.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000

Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

EXAMPLE

a(6) = 3 since 6 = 3 + p(1) + p(2) = 2 + p(1) + p(3) = 2 + p(2) + p(2) with 2 and 3 prime.

MATHEMATICA

PQ[n_]:=n>1&&PrimeQ[n]

a[n_]:=Sum[If[PQ[n-PartitionsP[m]-PartitionsP[k]], 1, 0], {m, 1, n}, {k, 1, m}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000040, A000041, A232398.

Sequence in context: A333995 A301764 A181833 * A228286 A158973 A071323

Adjacent sequences:  A202647 A202648 A202649 * A202651 A202652 A202653

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 24 2013

STATUS

approved

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Last modified June 28 21:04 EDT 2022. Contains 354907 sequences. (Running on oeis4.)