cp's OEIS Frontend

A237531 a(n) = |{0 < k < n/2: phi(k*(n-k)) - 1 and phi(k*(n-k)) + 1 are both prime}|, where phi(.) is Euler's totient function.

%I A237531 #8 Apr 06 2014 22:17:30
%S A237531 0,0,0,0,0,2,2,2,3,2,3,1,3,1,4,1,1,1,1,4,3,2,1,2,1,1,3,2,5,4,3,3,6,2,
%T A237531 3,1,5,4,5,2,3,5,5,3,7,6,3,7,3,8,7,4,4,5,6,4,3,9,6,8,9,8,9,9,10,7,6,3,
%U A237531 5,4,8,4,8,5,10,2,7,9,5,7
%N A237531 a(n) = |{0 < k < n/2: phi(k*(n-k)) - 1 and phi(k*(n-k)) + 1 are both prime}|, where phi(.) is Euler's totient function.
%C A237531 Conjecture: a(n) > 0 for all n > 5.
%C A237531 Clearly, this implies the twin prime conjecture.
%H A237531 Zhi-Wei Sun, <a href="/A237531/b237531.txt">Table of n, a(n) for n = 1..5000</a>
%H A237531 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A237531 a(12) = 1 since 12 = 3 + 9 with phi(3*9) - 1 = 17 and phi(3*9) + 1 = 19 both prime.
%e A237531 a(19) = 1 since 19 = 1 + 18 with phi(1*18) - 1 = 5 and phi(1*18) + 1 = 7 both prime.
%e A237531 a(86) = 1 since 86 = 8 + 78 with phi(8*78) - 1 = 191 and phi(8*78) + 1 = 193 both prime.
%t A237531 p[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
%t A237531 a[n_]:=Sum[If[p[k(n-k)],1,0],{k,1,(n-1)/2}]
%t A237531 Table[a[n],{n,1,80}]
%Y A237531 Cf. A000010, A000040, A001359, A006512, A014574, A072281, A233547, A234200, A237127, A237130, A237168, A237523.
%K A237531 nonn
%O A237531 1,6
%A A237531 _Zhi-Wei Sun_, Feb 09 2014