A261136 Primes p such that prime(p)-p+1 = prime(q) for some prime q.

3, 7, 71, 103, 173, 211, 271, 293, 1117, 1451, 1531, 1753, 1787, 1801, 2089, 2239, 2341, 2371, 2713, 2999, 3019, 3779, 3881, 3917, 4159, 4447, 4513, 4591, 4969, 5107, 5483, 5573, 5591, 5701, 5813, 5867, 6011, 6271, 6311, 6361, 6397, 6427, 7243, 8467, 8513, 9157, 9343, 9433, 9719, 10103

Offset

1,1

Comments

The conjecture in A260753 implies that the current sequence has infinitely many terms.

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..10000

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

Example

a(1) = 3 since prime(3)-3+1 = 5-3+1 = prime(2) with 3 and 2 both prime.
a(3) = 71 since prime(71)-71+1 = 353-70 = 283 = prime(61) with 71 and 61 both prime.

Mathematica

f[n_]:=Prime[Prime[n]]-Prime[n]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0; Do[If[PQ[f[k]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 1241}]
prQ[x_]:=Module[{c=Prime[x]-x+1}, AllTrue[{c, PrimePi[c]}, PrimeQ]]; Select[Prime[ Range[ 2000]], prQ] (* Harvey P. Dale, Apr 27 2023 *)

Crossrefs

Cf. A000040, A234695, A260753, A261361.

Sequence in context: A165589 A184316 A127177 * A127179 A359748 A113841

Adjacent sequences: A261133 A261134 A261135 * A261137 A261138 A261139

Keyword

nonn

Author

Zhi-Wei Sun, Aug 18 2015

Status

approved