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A095389 a(n) is the number of residues from reduced residue system, R, modulo 210 such that both R and R+2 are primes, i.e., both 210n+r and 210n+r+2 are primes at fixed n.

%I A095389 #17 Jul 08 2018 01:24:45
%S A095389 13,7,6,5,5,4,6,5,5,6,6,2,6,2,3,6,7,3,4,6,6,4,5,4,2,3,6,4,1,4,2,5,5,3,
%T A095389 4,4,2,2,2,4,3,2,5,2,5,2,4,4,3,5,2,2,4,2,3,2,4,4,3,1,1,4,1,2,0,6,5,2,
%U A095389 3,4,1,0,4,1,5,1,4,3,1,3,3,3,3,3,5,7,3,2,2,0,3,3,4,2,3,4,2,4,4,3,4,2,6,3,1
%N A095389 a(n) is the number of residues from reduced residue system, R, modulo 210 such that both R and R+2 are primes, i.e., both 210n+r and 210n+r+2 are primes at fixed n.
%C A095389 Since arbitrarily large prime gaps occur, several consecutive zeros may arise in the sequence.
%H A095389 Michael De Vlieger, <a href="/A095389/b095389.txt">Table of n, a(n) for n = 0..10000</a>
%e A095389 n=0: only 13+2=15 integers correspond to the condition: {11,17,29,41,59,71,101,107,137,149,179,191,197}, so a[0]=13; see A078859.
%e A095389 n=11: only 2 twins were found, {2339,2341} and {2381,2383} corresponding to residue pairs {29,31} and {71,73}.
%t A095389 {k =0, ta=Table[0, {100}]}; Do[{m=0};Do[s=210k+r;s1=210k+r+2; If[PrimeQ[s]&&PrimeQ[s+2], m=m+1], {r, 1, 210}];ta[[k]]=m, {k, 1, 100}];ta
%t A095389 (* Second program: *)
%t A095389 With[{P = Product[Prime@ i, {i, 4}]}, Function[R, Array[Count[R + P #, k_ /; Times @@ Boole@ PrimeQ@ {k, k + 2} == 1] &, 105, 0]]@ Select[Partition[Select[Range[P + 1], CoprimeQ[#, P] &], 2, 1], Differences@ # == {2} &][[All, 1]]] (* _Michael De Vlieger_, May 15 2017 *)
%Y A095389 Cf. A001359, A078859.
%K A095389 nonn
%O A095389 0,1
%A A095389 _Labos Elemer_ and _Enoch Haga_, Jun 15 2004