A236470 - OEIS

%I #7 Jan 03 2024 07:42:06

%S 0,1,1,0,1,1,0,1,0,0,1,1,0,2,1,2,1,1,1,1,0,2,2,2,0,1,0,1,1,0,3,0,1,0,

%T 1,3,0,1,1,1,0,1,0,0,1,2,1,0,0,0,0,2,0,1,0,0,0,0,1,1,0,0,2,1,0,1,1,0,

%U 0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0

%N a(n) = |{0 < k < n: p = prime(k) + phi(n-k), p + 2 and prime(p) + 2 are all prime}|, where phi(.) is Euler's totient function.

%C Conjecture: a(n) > 0 for all n > 948.

%C We have verified this for n up to 50000.

%C The conjecture implies that there are infinitely many primes p with p + 2 and prime(p) + 2 both prime. See A236458 for such primes p.

%H Zhi-Wei Sun, <a href="/A236470/b236470.txt">Table of n, a(n) for n = 1..10000</a>

%e  a(12) = 1 since prime(5) + phi(7) = 11 + 6 = 17, 17 + 2 = 19 and prime(17) + 2 = 59 + 2 = 61 are all prime.

%e a(97) = 1 since prime(7) + phi(90) = 17 + 24 = 41, 41 + 2 = 43 and prime(41) + 2 = 179 + 2 = 181 are all prime.

%t p[n_]:=PrimeQ[n]&&PrimeQ[n+2]&&PrimeQ[Prime[n]+2]

%t f[n_,k_]:=Prime[k]+EulerPhi[n-k]

%t a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000010, A000040, A001359, A006512, A236097, A236456, A236458, A236468.

%K nonn

%O 1,14

%A _Zhi-Wei Sun_, Jan 26 2014