

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
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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



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[nPartitionsP[m]PartitionsP[k]], 1, 0], {m, 1, n}, {k, 1, m}]
Table[a[n], {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



