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%I #19 Jan 26 2019 14:27:07
%S 1,1,1,2,2,4,2,2,1,1,1,3,2,2,2,3,2,10,4,1,4,5,7,4,4,15,1,1,1,2,19,15,
%T 4,8,13,4,4,10,2,4,1,4,15,16,6,3,5,5,10,6,7,4,20,10,4,1,6,13,3,1,14,4,
%U 25,8,21,39,29,8,2,14,1,34,16,12,17
%N Least k > 0 such that p(n)+k is prime, where p(n) is the number of partitions of n.
%C Conjecture of Zhi-Wei Sun: a(n) <= n for n > 0.
%H Sean A. Irvine, <a href="/A239675/b239675.txt">Table of n, a(n) for n = 0..9999</a>
%H 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(i).
%e a(3)=2 because p(3)=3 and p(3)+1=4 is composite, but p(3)+2=5 is prime.
%t a[n_] := a[n] = For[pn = PartitionsP[n]; k = 1, True, k++, If[PrimeQ[pn+k], Return[k]]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jan 26 2019 *)
%o (PARI) s=[]; for(n=0, 100, k=1; while(!isprime(numbpart(n)+k), k++); s=concat(s, k)); s \\ _Colin Barker_, Mar 26 2014
%Y Cf. A000009, A000040, A000041, A238457, A239736, A240545.
%K nonn
%O 0,4
%A _Sean A. Irvine_, Mar 23 2014