A260882 Least prime p such that 2*prime(p*n)+1 = prime(q*n) for some prime q.

3, 47, 3, 13, 797, 89, 2269, 733, 7877, 53, 14683, 16267, 17167, 59951, 10067, 761, 94463, 12437, 124561, 71881, 52009, 6791, 10061, 47287, 10789, 19009, 4813, 23173, 27427, 18701, 23011, 44917, 17, 70937, 883, 727, 99079, 10531, 18749, 126541, 18121, 34807, 29873, 159473, 853, 165317, 80627, 159721, 8263, 411707

Offset

1,1

Comments

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

This is a supplement to the conjecture in A260120. It implies that there are infinitely many Sophie Germain primes.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Example

a(2) = 47 since 2*prime(47*2)+1 = 2*491+1 = 983 = prime(83*2) with 47 and 83 both prime.
a(199) = 2784167 since 2*prime(2784167*199)+1 = 2*12290086499+1 = 24580172999 = prime(5399231*199) with 2784167 and 5399231 both prime.

Mathematica

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

Crossrefs

Cf. A000040, A005384, A260120, A260252.

Sequence in context: A339177 A000576 A336829 * A128109 A159246 A178466

Adjacent sequences: A260879 A260880 A260881 * A260883 A260884 A260885

Keyword

nonn

Author

Zhi-Wei Sun, Aug 02 2015

Status

approved