A236389 - OEIS

%I #7 Jan 24 2014 19:17:36

%S 0,0,0,0,0,0,0,1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,

%T 0,0,0,2,0,0,0,1,1,0,2,2,1,0,0,2,1,0,4,3,2,0,2,3,2,2,4,2,2,1,3,2,1,2,

%U 3,3,5,3,3,4,2,8,3,2,4,4,2,4,3,5,3,5,5,3,7,3,6,6,6,4,4,2,9,3,5,5

%N a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2^m*p(m) + 1 prime}|, where p(.) is the partition function (A000041).

%C Conjecture: a(n) > 0 for all n > 56.

%C We have verified this for n up to 33000.

%C The conjecture implies that there are infinitely many positive integers m with 2^m*p(m) + 1 prime. See A236390 for a list of such numbers m.

%H Zhi-Wei Sun, <a href="/A236389/b236389.txt">Table of n, a(n) for n = 1..10000</a>

%e a(10) = 1 since phi(1)/2 + phi(9)/12 = 1/2 + 6/12 = 1 with 2^1*p(1) + 1 = 2 + 1 = 3 prime.

%e a(30) = 1 since phi(17)/2 + phi(13)/12 = 8 + 1 = 9 with 2^9*p(9) + 1 = 512*30 + 1 = 15361 prime.

%e a(8261) = 1 since phi(395)/2 + phi(8261-395)/12 = 156 + 198 = 354 with 2^(354)*p(354) + 1 = 2^(354)*363117512048110005 + 1 prime.

%t q[n_]:=IntegerQ[n]&&PrimeQ[2^n*PartitionsP[n]+1]

%t f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12

%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, A000041, A000079, A236390.

%K nonn

%O 1,38

%A _Zhi-Wei Sun_, Jan 24 2014