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a(n) = |{0 < k <= n: p(n) + k is prime}|, where p(.) is the partition function (A000041).
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%I #6 Feb 27 2014 03:43:31

%S 1,1,1,1,1,2,2,2,2,2,3,3,4,3,3,3,3,3,4,4,3,4,2,3,2,4,4,4,3,1,4,4,4,3,

%T 3,4,4,5,4,5,4,3,3,5,2,2,8,6,6,2,4,5,6,3,7,6,4,6,5,6,4,3,3,4,2,4,5,7,

%U 5,6,4,7,7,5,2,5,6,2,6,5,4,7,7,5,6,5,3,6,2,6,4,9,8,2,5,7,6,4,2,8

%N a(n) = |{0 < k <= n: p(n) + k is prime}|, where p(.) is the partition function (A000041).

%C Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 3, 4, 5, 30, 109. Also, for each n > 2 there is a positive integer k <= n+3 such that p(n) - k is prime.

%C (ii) For the strict partition function q(.) given by A000009, we have |{0 < k <= n: q(n) + k is prime}| > 0 for all n > 0 and |{0 < k <= n: q(n) - k is prime}| > 0 for all n > 4.

%H Zhi-Wei Sun, <a href="/A238457/b238457.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014.

%e a(5) = 1 since p(5) + 4 = 7 + 4 = 11 is prime.

%e a(30) = 1 since p(30) + 19 = 5604 + 19 = 5623 is prime.

%e a(109) = 1 since p(109) + 63 = 541946240 + 63 = 541946303 is prime.

%t p[n_,k_]:=PrimeQ[PartitionsP[n]+k]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n}]

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

%Y Cf. A000009, A000040, A000041, A185636.

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Feb 27 2014