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A236548 Number of ways to write n = k^2 + m with k > 0 and m > 0 such that phi(k^2) + prime(m) is prime, where phi(.) is Euler's totient function.

%I A236548 #9 Jan 28 2014 15:25:51
%S A236548 0,1,0,0,0,1,1,0,1,0,1,1,1,2,1,1,1,2,1,1,3,1,0,2,2,1,2,1,0,3,0,4,1,2,
%T A236548 1,2,2,1,2,3,3,2,1,1,2,2,2,3,4,0,2,2,4,2,1,4,3,2,2,3,3,2,1,3,2,2,5,4,
%U A236548 3,1,0,3,4,2,1,1,3,3,2,2,2,3,3,3,5,2,3,3,2,1,4,2,4,3,4,4,3,3,3,0
%N A236548 Number of ways to write n = k^2 + m with k > 0 and m > 0 such that phi(k^2) + prime(m) is prime, where phi(.) is Euler's totient function.
%C A236548 Conjecture: (i) a(n) > 0 for all n > 100.
%C A236548 (ii) For any integer n > 4, there is a positive integer k < n with phi(k)^2 + prime(n-k)^2 prime.
%C A236548 Note that phi(k^2) (k = 1, 2, 3, ...) are pairwise distinct and this can be easily proved by induction.
%H A236548 Zhi-Wei Sun, <a href="/A236548/b236548.txt">Table of n, a(n) for n = 1..10000</a>
%e A236548 a(16) = 1 since 16 = 3^2 + 7 with phi(3^2) + prime(7) = 6 + 17 = 23 prime.
%e A236548 a(611) = 1 since 611 = 22^2 + 127 with phi(22^2) + prime(127) = 220 + 709 = 929 prime.
%t A236548 p[n_,k_]:=PrimeQ[EulerPhi[k^2]+Prime[n-k^2]]
%t A236548 a[n_]:=Sum[If[p[n,k],1,0],{k,1,Sqrt[n-1]}]
%t A236548 Table[a[n],{n,1,100}]
%Y A236548 Cf. A000010, A000040, A233867, A233918, A234246, A236541.
%K A236548 nonn
%O A236548 1,14
%A A236548 _Zhi-Wei Sun_, Jan 28 2014