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%I #6 Feb 02 2014 05:59:24
%S 0,1,2,2,1,2,2,3,3,1,4,4,3,5,3,1,1,4,5,6,3,1,4,4,3,2,2,3,3,5,3,6,5,1,
%T 6,1,4,6,4,1,6,7,8,6,2,2,5,8,4,4,3,3,7,8,3,5,3,4,6,7,8,9,5,2,3,2,4,7,
%U 5,2,2,6,6,8,5,1,6,2,6,7,3,3,8,8,6,5,2,5,6,9,9,5,4,1,7,2,3,9,6,3
%N Number of ordered ways to write n = k + m with k > 0 and m > 0 such that 6*k - 1, 6*k + 1 and k + phi(m) are all prime, where phi(.) is Euler's totient function.
%C Conjecture: (i) a(n) > 0 for all n > 1. Also, any n > 12 can be written as k + m (k > 0 and m > 2) with 6*k - 1, 6*k + 1 and k + phi(m)/2 all prime.
%C (ii) Each integer n > 34 can be written as p + q (q > 0) with p and p + phi(q) both prime. Also, any integer n > 14 can be written as p + q (q > 2) with p, p + 6 and p + phi(q)/2 all prime.
%C Clearly, part (i) of the conjecture implies that any integer n > 1 can be written as p + m - phi(m), where p is a prime and m is a positive integer.
%H Zhi-Wei Sun, <a href="/A236968/b236968.txt">Table of n, a(n) for n = 1..10000</a>
%e a(17) = 1 since 17 = 7 + 10 with 6*7 - 1 = 41, 6*7 + 1 = 43 and 7 + phi(10) = 7 + 4 = 11 all prime.
%e a(486) = 1 since 486 = 325 + 161 with 6*325 - 1 = 1949, 6*325 + 1 = 1951 and 325 + phi(161) = 325 + 132 = 457 all prime.
%t p[n_,k_]:=PrimeQ[6k-1]&&PrimeQ[6k+1]&&PrimeQ[k+EulerPhi[n-k]]
%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000010, A000040, A001359, A002822, A006512, A182662, A199920, A236531, A236831.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Feb 02 2014
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