cp's OEIS Frontend

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

%I A261136 #19 Apr 27 2023 15:28:27
%S A261136 3,7,71,103,173,211,271,293,1117,1451,1531,1753,1787,1801,2089,2239,
%T A261136 2341,2371,2713,2999,3019,3779,3881,3917,4159,4447,4513,4591,4969,
%U A261136 5107,5483,5573,5591,5701,5813,5867,6011,6271,6311,6361,6397,6427,7243,8467,8513,9157,9343,9433,9719,10103
%N A261136 Primes p such that prime(p)-p+1 = prime(q) for some prime q.
%C A261136 The conjecture in A260753 implies that the current sequence has infinitely many terms.
%D A261136 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.
%H A261136 Zhi-Wei Sun, <a href="/A261136/b261136.txt">Table of n, a(n) for n = 1..10000</a>
%H A261136 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A261136 a(1) = 3 since prime(3)-3+1 = 5-3+1 = prime(2) with 3 and 2 both prime.
%e A261136 a(3) = 71 since prime(71)-71+1 = 353-70 = 283 = prime(61) with 71 and 61 both prime.
%t A261136 f[n_]:=Prime[Prime[n]]-Prime[n]+1
%t A261136 PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
%t A261136 n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1241}]
%t A261136 prQ[x_]:=Module[{c=Prime[x]-x+1},AllTrue[{c,PrimePi[c]},PrimeQ]]; Select[Prime[ Range[ 2000]],prQ] (* _Harvey P. Dale_, Apr 27 2023 *)
%Y A261136 Cf. A000040, A234695, A260753, A261361.
%K A261136 nonn
%O A261136 1,1
%A A261136 _Zhi-Wei Sun_, Aug 18 2015