cp's OEIS Frontend

A236573 Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2*k + phi(m) - 1, prime(p + 2) + 2 and 2*n - p are all prime, where phi(.) is Euler's totient function.

%I A236573 #6 Jan 29 2014 10:33:34
%S A236573 0,0,0,1,2,2,1,2,3,0,2,3,0,1,0,0,2,1,2,0,2,3,1,4,3,3,8,3,2,5,5,4,3,1,
%T A236573 2,7,6,0,8,4,2,8,4,4,7,4,4,3,6,3,5,3,1,4,6,4,9,2,4,11,2,1,5,2,4,4,1,2,
%U A236573 9,4,0,3,2,2,5,2,4,4,1,4,1,1,1,4,0,0,3,2,5,5,0,1,2,2,1,2,1,2,2,1
%N A236573 Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2*k + phi(m) - 1, prime(p + 2) + 2 and 2*n - p are all prime, where phi(.) is Euler's totient function.
%C A236573 Conjecture: a(n) > 0 for all n > 712.
%C A236573 This implies the conjecture in A236566.
%H A236573 Zhi-Wei Sun, <a href="/A236573/b236573.txt">Table of n, a(n) for n = 1..10000</a>
%e A236573 a(100) = 1 since 100 = 10 + 90 with 2*10 + phi(90) - 1 = 20 + 24 - 1 = 43, prime(43 + 2) + 2 = 197 + 2 = 199 and 2*100 - 43 = 157 all prime.
%e A236573 a(1727) = 1 since 1727 = 956 + 771 with 2*956 + phi(771) - 1 = 1912 + 512 - 1 = 2423, prime(2423 + 2) + 2 = 21599 + 2 = 21601 and 2*1727 - 2423 = 1031 all prime.
%t A236573 p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n+2]+2]
%t A236573 f[n_,k_]:=2k+EulerPhi[n-k]-1
%t A236573 a[n_]:=Sum[If[p[f[n,k]]&&PrimeQ[2n-f[n,k]],1,0],{k,1,n-1}]
%t A236573 Table[a[n],{n,1,100}]
%Y A236573 Cf. A000010, A000040, A001359, A002372, A002375, A006512, A234808, A236531, A236566, A236568, A236569.
%K A236573 nonn
%O A236573 1,5
%A A236573 _Zhi-Wei Sun_, Jan 29 2014