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%I #12 Jun 24 2017 01:02:38
%S 0,0,0,0,0,0,0,1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,2,
%U 1,0,0,0,0,1,0,1,1,0,0,0,1,0,2,1,0,1,0,2,0,1,1,0,0,4,1,2,1,0,2,3
%N a(n) = |{0 < k < n: r = phi(k) + phi(n-k)/6 + 1 and p(r) + q(r) are both prime}|, where phi(.) is Euler's totient function, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).
%C Conjecture: a(n) > 0 for all n > 127.
%C We have verified this for n up to 30000.
%C The conjecture implies that there are infinitely many primes r with p(r) + q(r) also prime.
%H Zhi-Wei Sun, <a href="/A236419/b236419.txt">Table of n, a(n) for n = 1..10000</a>
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e a(15) = 1 since phi(1) + phi(14)/6 + 1 = 3 with p(3) + q(3) = 3 + 2 = 5 prime.
%e a(54) = 1 since phi(41) + phi(13)/6 + 1 = 43 with p(43) + q(43) = 63261 + 1610 = 64871 prime.
%t pq[n_]:=PrimeQ[n]&&PrimeQ[PartitionsP[n]+PartitionsQ[n]]
%t f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/6+1
%t a[n_]:=Sum[If[pq[f[n,k]],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000009, A000010, A000040, A232504, A236412, A236417.
%K nonn
%O 1,68
%A _Zhi-Wei Sun_, Jan 25 2014
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