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

2, 2, 2, 47, 1481, 31, 11, 557, 277, 1847, 7, 3, 1861, 47, 1451, 557, 1429, 2, 18367, 2069, 13411, 463, 26731, 7, 50119, 61, 101, 877, 29, 11261, 2971, 421, 298589, 32633, 31, 55933, 5521, 7307, 22349, 11, 641, 13, 47881, 3, 2309, 51673, 94309, 186679, 136207, 1301

Offset: 1

Author

Zhi-Wei Sun, Jul 12 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0.

This implies the conjecture that the sequence p(n) (n = 1,2,3,...) contains infinitely many primes.

Examples

			a(1) = 2 since p(2*1) = 2 is prime.
a(4) = 47 since 47 and p(47*4) = p(188) = 1398341745571 are both prime.
		

References

• 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.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..88
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000041, A046063, A049575, A259764.

Programs

Mathematica

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}]

PARI

a(n)={my(r=1); while(!isprime(numbpart(prime(r)*n)), r++); return(prime(r));}
main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 12 2015 */