cp's OEIS Frontend

A233417 a(n) = |{0 < k <= n/2: q(k)*q(n-k) + 1 is prime}|, where q(.) is the strict partition function (A000009).

%I A233417 #17 Aug 05 2019 09:27:47
%S A233417 0,1,1,2,2,2,2,3,2,2,4,5,3,1,5,7,1,3,4,4,3,2,5,3,6,6,1,6,8,6,6,4,7,7,
%T A233417 3,5,5,6,6,5,5,3,7,8,7,7,8,8,6,4,8,8,5,3,8,8,5,15,6,8,3,9,5,6,7,9,4,6,
%U A233417 8,9,5,4,7,8,7,6,10,9,9,8,6,6,9,9,7,12,5,10,7,7,5,3,8,10,7,5,9,7,4,5
%N A233417 a(n) = |{0 < k <= n/2: q(k)*q(n-k) + 1 is prime}|, where q(.) is the strict partition function (A000009).
%C A233417 Conjecture: (i) a(n) > 0 for all n > 1. Similarly, for any integer n > 5, there is a positive integer k < n with q(k)*q(n-k) - 1 prime.
%C A233417 (ii) Let n > 1 be an integer. Then p(k) + q(n-k)^2 is prime for some 0 < k < n, where p(.) is the partition function (A000041). If n is not equal to 8, then k^3 + q(n-k)^2 is prime for some 0 < k < n.
%H A233417 Zhi-Wei Sun, <a href="/A233417/b233417.txt">Table of n, a(n) for n = 1..10000</a>
%H A233417 Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
%e A233417 a(14) = 1 since q(1)*q(13) + 1 = 1*18 + 1 = 19 is prime.
%e A233417 a(17) = 1 since q(4)*q(13) + 1 = 2*18 + 1 = 37 is prime.
%e A233417 a(27) = 1 since q(13)*q(14) + 1 = 18*22 + 1 = 397 is prime.
%t A233417 a[n_]:=Sum[If[PrimeQ[PartitionsQ[k]*PartitionsQ[n-k]+1],1,0],{k,1,n/2}]
%t A233417 Table[a[n],{n,1,100}]
%Y A233417 Cf. A000009, A000040, A232504, A233307, A233346, A233359, A233390, A233393.
%K A233417 nonn
%O A233417 1,4
%A A233417 _Zhi-Wei Sun_, Dec 09 2013