A236374 - OEIS

%I #7 Jan 24 2014 10:24:12

%S 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 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 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 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 Conjecture: a(n) > 0 for all n > 31.

%C We have verified this for n up to 60000.

%C The conjecture implies that there are infinitely many positive integers m with 2^(m-1)*phi(m) - 1 prime.

%C See A236375 for a list of known numbers m with 2^(m-1)*phi(m) - 1 prime.

%H Zhi-Wei Sun, <a href="/A236374/b236374.txt">Table of n, a(n) for n = 1..10000</a>

%e a(24) = 1 since phi(8)/2 + phi(16)/8 = 3 with 2^(3-1)*phi(3) - 1 = 7 prime.

%e a(33) = 1 since phi(13)/2 + phi(20)/8 = 7 with 2^(7-1)*phi(7) - 1 = 383 prime.

%e 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 q[n_]:=IntegerQ[n]&&PrimeQ[2^(n-1)*EulerPhi[n]-1]

%t f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8

%t a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000010, A000040, A000079, A236375.

%K nonn

%O 1,20

%A _Zhi-Wei Sun_, Jan 24 2014