%I #16 Sep 08 2022 08:45:35
%S 1308498,3042492,3042702,7445310,20031102,31572522,44687988,54266292,
%T 141208620,182316522,237416370,357080022,448436322,611641188,
%U 699458412,761126028,774997368,794065968,836452962,915215592,944958942,1009194618,1581935940,1763255562,1871007372
%N Numbers k such that k, k + 210*1, k + 210*2, k + 210*3, k + 210*4 are averages of pairs of twin primes.
%e 1308498 and 1308708, 1308918, 1309128, 1309338 are averages of twin primes.
%e 3042492 and 3042702, 3042912, 3043122, 3043332 are averages of twin primes.
%t q=210; lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[n+q*1-1]&& PrimeQ[n+q*1+1]&& PrimeQ[n+q*2-1]&&PrimeQ[n+q*2+1]&& PrimeQ[n+q*3-1]&& PrimeQ[n+q*3+1]&& PrimeQ[n+q*4-1]&&PrimeQ[n+q*4+1],AppendTo[lst,n]],{n,10^8}]; lst
%t Select[Range[55*10^6],AllTrue[{#-1,#+1,#+209,#+211,#+419,#+421,#+629,#+631,#+839,#+841},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Aug 08 2017 *)
%o (Magma) [k:k in [1..55000000]|forall{m:m in [0,210,420,630,840]|IsPrime(k+m-1) and IsPrime(k+m+1)}]; // _Marius A. Burtea_, Jan 13 2020
%Y Cf. A001097, A001359, A006512, A014574.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Aug 20 2008
%E Edited by _N. J. A. Sloane_, Aug 24 2008
%E a(9)-a(25) from _Amiram Eldar_, Jan 13 2020