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.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Jan 13 2014

Keywords

• nonn

Comments

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

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

Examples

			a(14) = 1 since 14 = 4 + 10 with prime(4) + phi(10) = 11 and 11*12 - prime(11) = 101 both prime.
a(15) = 1 since 15 = 6 + 9 with prime(6) + phi(9) = 19 and 19*20 - prime(19) = 313 both prime.
a(37) = 1 since 37 = 23 + 14 with prime(23) + phi(14) = 89 and 89*90 - prime(89) = 7549 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Crossrefs

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

Programs

Mathematica

PQ[n_]:=PrimeQ[n]&&PrimeQ[n(n+1)-Prime[n]]
f[n_,k_]:=Prime[k]+EulerPhi[n-k]
a[n_]:=Sum[If[PQ[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]