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A239936 Least k > 0 such that p(k)+q(n) is prime, where p(n) is the number of partitions of n and q(n) is the number of strict partitions of n.

%I A239936 #11 Feb 19 2019 04:06:35
%S A239936 1,1,1,1,1,2,1,2,1,3,1,1,2,1,1,2,4,3,1,4,3,3,10,3,4,5,2,1,1,1,6,5,5,1,
%T A239936 6,2,4,1,12,1,15,13,1,3,5,4,4,1,5,5,1,2,1,12,49,1,1,2,6,6,3,14,3,3,3,
%U A239936 6,6,16,13,16,11,1,1,4,5,3,12,25
%N A239936 Least k > 0 such that p(k)+q(n) is prime, where p(n) is the number of partitions of n and q(n) is the number of strict partitions of n.
%C A239936 Conjecture of Zhi-Wei Sun: a(n) < n for n > 1.
%H A239936 Sean A. Irvine, <a href="/A239936/b239936.txt">Table of n, a(n) for n = 0..9999</a>
%H A239936 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014-2016. See Conjecture 4.1(ii).
%e A239936 a(5)=2 since q(5)+p(2)=3+2=5 is prime but q(5)+p(1)=4 is composite.
%t A239936 a[n_] := For[k = 1, True, k++, If[PrimeQ[PartitionsP[k] + PartitionsQ[n]], Return[k]]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 19 2019 *)
%Y A239936 Cf. A000009, A000040, A000041, A239675, A239736.
%K A239936 nonn
%O A239936 0,6
%A A239936 _Sean A. Irvine_, Mar 29 2014