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

1,1

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.

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.

Example

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.

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}]

Prog

(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 */

Crossrefs

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

Sequence in context: A254131 A339014 A175910 * A075042 A045983 A266713

Adjacent sequences: A257659 A257660 A257661 * A257663 A257664 A257665

Keyword

nonn

Author

Zhi-Wei Sun, Jul 12 2015

Status

approved