cp's OEIS Frontend

A233539 a(n) = |{0 < k < n-2: m - 1, m + 1, prime(m) - m and prime(m) + m are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

%I A233539 #32 Jan 13 2014 21:59:28
%S A233539 0,0,0,0,0,0,0,2,2,2,3,2,4,2,4,4,2,4,3,5,1,4,2,3,1,2,2,2,1,1,0,0,1,4,
%T A233539 0,1,2,0,5,2,4,4,1,3,3,3,2,3,8,2,2,3,5,5,4,3,5,3,4,3,1,3,8,4,5,4,2,6,
%U A233539 0,12,2,4,1,5,0,4,1,4,3,3,2,5,4,7,5,3,11,1,5,4,3,4,6,2,2,5,5,6,4,4
%N A233539 a(n) = |{0 < k < n-2: m - 1, m + 1, prime(m) - m and prime(m) + m are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.
%C A233539 Conjecture: (i) a(n) > 0 for all n > 794.
%C A233539 (ii) For any integer n > 59, there is a positive integer k < n such that m = phi(k) + phi(n-k)/4 is an integer with prime(m) - m and prime(m) + m both prime.
%C A233539 Clearly, part (i) of the conjecture implies that there are infinitely many positive integers m with m - 1, m + 1, prime(m) - m and prime(m) + m all prime.
%H A233539 Zhi-Wei Sun, <a href="/A233539/b233539.txt">Table of n, a(n) for n = 1..10000</a>
%e A233539 a(21) = 1 since phi(6) + phi(15)/2 = 6 with 6 - 1 = 5, 6 + 1 = 7, prime(6) - 6 = 7 and prime(6) + 6 = 19 all prime.
%e A233539 a(25) = 1 since phi(17) + phi(8)/2 = 18 with 18 - 1 = 17, 18 + 1 = 19, prime(18) - 18 = 43 and prime(18) + 18 = 79 all prime.
%t A233539 q[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]-n]&&PrimeQ[Prime[n]+n]
%t A233539 f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2
%t A233539 a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-3}]
%t A233539 Table[a[n],{n,1,100}]
%Y A233539 Cf. A000010, A000040, A014574, A064269, A064402, A232861, A235682.
%K A233539 nonn
%O A233539 1,8
%A A233539 _Zhi-Wei Sun_, Jan 13 2014