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%I #19 Dec 27 2013 12:03:21
%S 0,0,0,1,2,2,1,1,1,0,1,0,2,2,2,3,4,2,4,2,3,3,3,2,2,3,1,4,2,1,4,2,4,2,
%T 5,3,4,1,5,6,4,2,5,5,5,3,5,4,6,3,5,7,10,2,4,5,6,5,5,2,3,5,6,6,4,2,5,3,
%U 7,4,5,3,8,7,2,5,9,3,3,2,9,9,6,6,7,6,9,4,7,4,10,8,6,11,11,4,6,4,9,7
%N Number of ways to write n = k + m with k > 0 and m > 0 such that p = k + phi(m)/2 and q(p) + 1 are both prime, where phi(.) is Euler's totient function, and q(.) is the strict partition function (A000009).
%C Conjecture: (i) a(n) > 0 for all n > 12.
%C (ii) For any integer n > 4, there is a prime p < n - 2 such that q(p + phi(n-p)/2) + 1 is prime.
%C Clearly, part (i) of the conjecture implies that there are infinitely many primes p with q(p) + 1 prime (cf. A234530).
%C We have verified part (i) for n up to 10^5.
%H Zhi-Wei Sun, <a href="/A234514/b234514.txt">Table of n, a(n) for n = 1..10000</a>
%e a(11) = 1 since 11 = 1 + 10 with 1 + phi(10)/2 = 3 and q(3) + 1 = 3 both prime.
%e a(27) = 1 since 27 = 7 + 20 with 7 + phi(20)/2 = 11 and q(11) + 1 = 13 both prime.
%e a(30) = 1 since 30 = 8 + 22 with 8 + phi(22)/2 = 13 and q(13) + 1 = 19 both prime.
%e a(38) = 1 since 38 = 21 + 17 with 21 + phi(17)/2 = 29 and q(29) + 1 = 257 both prime.
%e a(572) = 1 since 572 = 77 + 495 with 77 + phi(495)/2 = 197 and q(197) + 1 = 406072423 both prime.
%e a(860) = 1 since 860 = 523 + 337 with 523 + phi(337)/2 = 691 and q(691) + 1 = 712827068077888961 both prime.
%t f[n_,k_]:=k+EulerPhi[n-k]/2
%t q[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[PartitionsQ[f[n,k]]+1]
%t a[n_]:=Sum[If[q[n,k],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000009, A000010, A000040, A229835, A233307, A233390, A233417, A234451, A234470, A234475, A234503, A234504, A234530
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Dec 27 2013
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