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

Author

Zhi-Wei Sun, Aug 18 2015

Keywords

• nonn

Comments

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

Examples

			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.
		

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.

Crossrefs

Cf. A000040, A234695, A260753, A261361.

Programs

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 *)