cp's OEIS Frontend

A236374 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with 2^(m-1)*phi(m) - 1 prime}|, where phi(.) is Euler's totient function.

%I A236374 #7 Jan 24 2014 10:24:12
%S A236374 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,0,2,1,2,2,2,4,2,1,0,2,1,2,
%T A236374 3,3,3,4,2,2,2,3,5,3,4,4,1,1,2,3,7,4,3,5,3,3,2,4,5,4,3,4,3,2,6,7,5,5,
%U A236374 4,4,5,4,5,5,3,7,3,5,1,7,4,7,7,5,9,5,9,3,3,5,13,7,9,7,3,4,10,10,9,11
%N A236374 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with 2^(m-1)*phi(m) - 1 prime}|, where phi(.) is Euler's totient function.
%C A236374 Conjecture: a(n) > 0 for all n > 31.
%C A236374 We have verified this for n up to 60000.
%C A236374 The conjecture implies that there are infinitely many positive integers m with 2^(m-1)*phi(m) - 1 prime.
%C A236374 See A236375 for a list of known numbers m with 2^(m-1)*phi(m) - 1 prime.
%H A236374 Zhi-Wei Sun, <a href="/A236374/b236374.txt">Table of n, a(n) for n = 1..10000</a>
%e A236374 a(24) = 1 since phi(8)/2 + phi(16)/8 = 3 with 2^(3-1)*phi(3) - 1 = 7 prime.
%e A236374 a(33) = 1 since phi(13)/2 + phi(20)/8 = 7 with 2^(7-1)*phi(7) - 1 = 383 prime.
%e A236374 a(79) = 1 since phi(27)/2 + phi(52)/8 = 9 + 3 = 12 with 2^(12-1)*phi(12) - 1 = 2^(13) - 1 = 8191 prime.
%t A236374 q[n_]:=IntegerQ[n]&&PrimeQ[2^(n-1)*EulerPhi[n]-1]
%t A236374 f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
%t A236374 a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]
%t A236374 Table[a[n],{n,1,100}]
%Y A236374 Cf. A000010, A000040, A000079, A236375.
%K A236374 nonn
%O A236374 1,20
%A A236374 _Zhi-Wei Sun_, Jan 24 2014