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A260882 Least prime p such that 2*prime(p*n)+1 = prime(q*n) for some prime q. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A307290 A193420 A000576 * A128109 A159246 A178466

Adjacent sequences:  A260879 A260880 A260881 * A260883 A260884 A260885

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Aug 02 2015

STATUS

approved

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Last modified October 21 17:14 EDT 2019. Contains 328302 sequences. (Running on oeis4.)