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A257662 Least prime q such that p(q*n) is prime, where p(.) is the partition function given by A000041.

%I A257662 #11 Jul 13 2015 04:27:49
%S A257662 2,2,2,47,1481,31,11,557,277,1847,7,3,1861,47,1451,557,1429,2,18367,
%T A257662 2069,13411,463,26731,7,50119,61,101,877,29,11261,2971,421,298589,
%U A257662 32633,31,55933,5521,7307,22349,11,641,13,47881,3,2309,51673,94309,186679,136207,1301
%N A257662 Least prime q such that p(q*n) is prime, where p(.) is the partition function given by A000041.
%C A257662 Conjecture: a(n) exists for any n > 0.
%C A257662 This implies the conjecture that the sequence p(n) (n = 1,2,3,...) contains infinitely many primes.
%D A257662 Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H A257662 Zhi-Wei Sun, <a href="/A257662/b257662.txt">Table of n, a(n) for n = 1..88</a>
%H A257662 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A257662 a(1) = 2 since p(2*1) = 2 is prime.
%e A257662 a(4) = 47 since 47 and p(47*4) = p(188) = 1398341745571 are both prime.
%t A257662 Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[Prime[k]*n]],Goto[aa],Goto[bb]]; Label[aa];Print[n, " ", Prime[k]];Continue,{n,1,50}]
%o A257662 (PARI) a(n)={my(r=1); while(!isprime(numbpart(prime(r)*n)), r++); return(prime(r));}
%o A257662 main(size)={return(vector(size,n,a(n)));} /* _Anders Hellström_, Jul 12 2015 */
%Y A257662 Cf. A000040, A000041, A046063, A049575, A259764.
%K A257662 nonn
%O A257662 1,1
%A A257662 _Zhi-Wei Sun_, Jul 12 2015