A236460 - OEIS

%I #11 Jan 19 2019 04:15:43

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

%T 1,2,2,2,1,1,2,0,4,1,2,1,5,2,1,3,1,1,3,2,6,3,0,2,5,5,6,3,4,5,3,4,4,4,

%U 6,3,2,6,2,3,2,10,2,3,1,6,1,4,0,2,3,4,2,4,0,4,0,3,2,3,0,4,0,1,1,4

%N Number of ordered ways to write n = k + m with k > 0 and m > 0 such that p = phi(k) + phi(m)/2 - 1, prime(p) + 4 and prime(p) + 6 are all prime, where phi(.) is Euler's totient function.

%C Conjecture: a(n) > 0 for all n > 211.

%C This implies that there are infinitely many primes p with {prime(p), prime(p) + 4, prime(p) + 6} a prime triple. See A236462 for such primes p.

%H Zhi-Wei Sun, <a href="/A236460/b236460.txt">Table of n, a(n) for n = 1..10000</a>

%e a(30) = 1 since 30 = 13 + 17 with phi(13) + phi(17)/2 - 1 = 19, prime(19) + 4 = 67 + 4 = 71 and prime(19) + 6 = 73 all prime.

%e a(831) = 1 since 831 = 66 + 765 with phi(66) + phi(765)/2 - 1 = 20 + 192 - 1 = 211, prime(211) + 4 = 1297 + 4 = 1301 and prime(211) + 6 = 1303 all prime.

%t p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]+4]&&PrimeQ[Prime[n]+6]

%t f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2-1

%t a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-3}]

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

%Y Cf. A000010, A000040, A001359, A006512, A022005, A236456, A236457, A236458, A236462, A236464.

%K nonn

%O 1,32

%A _Zhi-Wei Sun_, Jan 26 2014