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%I #11 Apr 05 2014 22:29:30
%S 0,0,0,0,0,1,1,2,2,3,3,4,4,3,4,5,5,4,7,7,6,5,5,7,3,6,7,7,5,7,4,8,4,7,
%T 7,8,7,4,5,5,4,4,5,5,6,5,4,5,3,5,4,6,6,4,6,5,4,3,6,4,9,4,8,6,7,6,8,4,
%U 7,4,7,8,9,2,3,1,8,6,9,6,6,6,6,4,7,5,8,8,4,5,5,9,7,10,4,10,3,7,8,6
%N Number of ways to write n = k + m with 2 < k <= m such that q(phi(k)*phi(m)/4) + 1 is prime, where phi(.) is Euler's totient function and q(.) is the strict partition function (A000009).
%C Conjecture: a(n) > 0 for all n > 5.
%C This implies that there are infinitely many primes p with p - 1 a term of A000009.
%H Zhi-Wei Sun, <a href="/A234475/b234475.txt">Table of n, a(n) for n = 1..2525</a>
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(6) = 1 since 6 = 3 + 3 with q(phi(3)*phi(3)/4) + 1 = q(1) + 1 = 2 prime.
%e a(76) = 1 since 76 = 18 + 58 with q(phi(18)*phi(58)/4) + 1 = q(42) + 1 = 1427 prime.
%e a(197) = 1 since 197 = 4 + 193 with q(phi(4)*phi(193)/4) + 1 = q(96) + 1 = 317789.
%e a(356) = 1 since 356 = 88 + 268 with q(phi(88)*phi(268)/4) + 1 = q(1320) + 1 = 35940172290335689735986241 prime.
%t f[n_,k_]:=PartitionsQ[EulerPhi[k]*EulerPhi[n-k]/4]+1
%t a[n_]:=Sum[If[PrimeQ[f[n,k]],1,0],{k,3,n/2}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000009, A000010, A000040, A232504, A233307, A233346, A233547, A233390, A233393, A234309, A234310, A234337, A234344, A234347, A234359, A234360, A234361, A234451, A234470, A234514, A234530, A234567, A234569
%K nonn
%O 1,8
%A _Zhi-Wei Sun_, Dec 26 2013
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