cp's OEIS Frontend

A232353 Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and p*(p+1) - prime(p) are both prime, where phi(.) is Euler's totient function.

%I A232353 #27 Jan 13 2014 11:11:36
%S A232353 0,1,1,0,1,1,0,1,1,2,1,2,2,1,1,2,4,3,2,6,3,4,4,5,5,3,4,2,6,5,4,6,5,4,
%T A232353 6,7,1,6,4,8,6,6,7,4,5,10,5,3,4,6,7,6,6,9,6,3,7,7,10,5,9,7,7,6,5,8,9,
%U A232353 4,6,9,8,5,8,5,8,8,5,6,7,9,10,8,8,8,11,10,11,7,8,13,9,6,12,10,5,9,7,8,14,8
%N A232353 Number of ways to write n = k + m with k > 0 and m > 0 such that p = prime(k) + phi(m) and p*(p+1) - prime(p) are both prime, where phi(.) is Euler's totient function.
%C A232353 Conjecture: a(n) > 0 for all n > 7.
%C A232353 This implies that there are infinitely many primes p with p*(p+1) - prime(p) prime.
%H A232353 Zhi-Wei Sun, <a href="/A232353/b232353.txt">Table of n, a(n) for n = 1..5000</a>
%e A232353 a(14) = 1 since 14 = 4 + 10 with prime(4) + phi(10) = 11 and 11*12 - prime(11) = 101 both prime.
%e A232353 a(15) = 1 since 15 = 6 + 9 with prime(6) + phi(9) = 19 and 19*20 - prime(19) = 313 both prime.
%e A232353 a(37) = 1 since 37 = 23 + 14 with prime(23) + phi(14) = 89 and 89*90 - prime(89) = 7549 both prime.
%t A232353 PQ[n_]:=PrimeQ[n]&&PrimeQ[n(n+1)-Prime[n]]
%t A232353 f[n_,k_]:=Prime[k]+EulerPhi[n-k]
%t A232353 a[n_]:=Sum[If[PQ[f[n,k]],1,0],{k,1,n-1}]
%t A232353 Table[a[n],{n,1,100}]
%Y A232353 Cf. A000010, A000040, A234694, A235592, A235613, A235614, A235661.
%K A232353 nonn
%O A232353 1,10
%A A232353 _Zhi-Wei Sun_, Jan 13 2014