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A259628 Numbers m with m-1, m+1, prime(m)+2, prime(m)-m, prime(m)+m, m*prime(m)-1 and m*prime(m)+1 all prime.

%I A259628 #18 Jan 26 2016 17:41:30
%S A259628 2523708,6740478,6759030,14655522,22885698,28384200,44630148,71742300,
%T A259628 87002328,87466500,89842200,153110622,153647490,184373490,283232040,
%U A259628 312124920,366318960,408770670,412216920,439429128,456486030,486730398,517602600,606159978,607942848,675661080,855983352,869593998,923864562,971400672
%N A259628 Numbers m with m-1, m+1, prime(m)+2, prime(m)-m, prime(m)+m, m*prime(m)-1 and m*prime(m)+1 all prime.
%C A259628 Conjecture: The sequence contains infinitely many terms.
%C A259628 This is stronger than the conjectures in A232861 and A259539.
%D A259628 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 A259628 Zhi-Wei Sun, <a href="/A259628/b259628.txt">Table of n, a(n) for n = 1..160</a>
%H A259628 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 A259628 a(1) = 2523708 since the seven numbers 2523707, 2523709, prime(2523708)+2 = 41578739+2 = 41578741, prime(2523708)-2523708 = 41578739-2523708 = 39055031, prime(2523708)+2523708 = 41578739+2523708 = 44102447, 2523708*prime(2523708)-1 = 2523708*41578739-1 = 104932596244211 and 2523708*prime(2523708)+1 = 2523708*41578739+1 = 104932596244213 are all prime.
%t A259628 TW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
%t A259628 n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]&&PrimeQ[Prime[Prime[k]+1]-Prime[k]-1]&&PrimeQ[Prime[Prime[k]+1]+Prime[k]+1]&&TW[(Prime[k]+1)Prime[Prime[k]+1]],n=n+1;Print[n," ",Prime[k]+1]],{k,1,5*10^7}]
%t A259628 allprQ[n_]:=Module[{p=Prime[n]},AllTrue[{n-1,n+1,p+2,p-n,p+n,n*p-1, n*p+1}, PrimeQ]]; Select[Range[98*10^7],allprQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Jan 26 2016 *)
%Y A259628 Cf. A000040, A014574, A232861, A259539.
%K A259628 nonn
%O A259628 1,1
%A A259628 _Zhi-Wei Sun_, Jul 01 2015