A232353 - OEIS

%I #27 Jan 13 2014 11:11:36

%S 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 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 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 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 Conjecture: a(n) > 0 for all n > 7.

%C This implies that there are infinitely many primes p with p*(p+1) - prime(p) prime.

%H Zhi-Wei Sun, <a href="/A232353/b232353.txt">Table of n, a(n) for n = 1..5000</a>

%e a(14) = 1 since 14 = 4 + 10 with prime(4) + phi(10) = 11 and 11*12 - prime(11) = 101 both prime.

%e a(15) = 1 since 15 = 6 + 9 with prime(6) + phi(9) = 19 and 19*20 - prime(19) = 313 both prime.

%e a(37) = 1 since 37 = 23 + 14 with prime(23) + phi(14) = 89 and 89*90 - prime(89) = 7549 both prime.

%t PQ[n_]:=PrimeQ[n]&&PrimeQ[n(n+1)-Prime[n]]

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

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

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

%Y Cf. A000010, A000040, A234694, A235592, A235613, A235614, A235661.

%K nonn

%O 1,10

%A _Zhi-Wei Sun_, Jan 13 2014