A260208 Least prime p such that 2p*n+1 = prime(q*n) for some prime q.

2, 3, 2, 107, 271, 3, 3, 523, 17, 191, 73, 2707, 587, 2017, 19, 233, 57193, 7583, 9791, 7, 2111, 1373, 43, 109, 1283, 463, 8179, 25583, 7489, 1733, 9011, 7753, 7853, 887, 10141, 71, 1373, 7927, 509, 1433, 4513, 2399, 4211, 26407, 307, 2843, 58579, 3121, 5519, 38371

Offset

1,1

Comments

Conjecture: a(n) exists for any n > 0. In general, if a > 0 is even and b is 1 or -1, then for any positive integer n there are primes p and q such that a*p*n+b = prime(q*n).

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..1000

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Example

a(5) = 271 since 2*271*5+1 = 2711 = prime(79*5) with 271 and 79 both prime.

Mathematica

PQ[n_, p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=0; Label[aa]; k=k+1; If[PQ[n, 2*Prime[k]*n+1]], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]

Crossrefs

Cf. A000040, A260197.

Sequence in context: A075108 A364139 A265590 * A242457 A340261 A190146

Adjacent sequences: A260205 A260206 A260207 * A260209 A260210 A260211

Keyword

nonn

Author

Zhi-Wei Sun, Jul 19 2015

Status

approved