A236573 Number of ordered ways to write n = k + m (k > 0, m > 0) such that p = 2*k + phi(m) - 1, prime(p + 2) + 2 and 2*n - p are all prime, where phi(.) is Euler's totient function.

0, 0, 0, 1, 2, 2, 1, 2, 3, 0, 2, 3, 0, 1, 0, 0, 2, 1, 2, 0, 2, 3, 1, 4, 3, 3, 8, 3, 2, 5, 5, 4, 3, 1, 2, 7, 6, 0, 8, 4, 2, 8, 4, 4, 7, 4, 4, 3, 6, 3, 5, 3, 1, 4, 6, 4, 9, 2, 4, 11, 2, 1, 5, 2, 4, 4, 1, 2, 9, 4, 0, 3, 2, 2, 5, 2, 4, 4, 1, 4, 1, 1, 1, 4, 0, 0, 3, 2, 5, 5, 0, 1, 2, 2, 1, 2, 1, 2, 2, 1

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 712.

This implies the conjecture in A236566.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(100) = 1 since 100 = 10 + 90 with 2*10 + phi(90) - 1 = 20 + 24 - 1 = 43, prime(43 + 2) + 2 = 197 + 2 = 199 and 2*100 - 43 = 157 all prime.
a(1727) = 1 since 1727 = 956 + 771 with 2*956 + phi(771) - 1 = 1912 + 512 - 1 = 2423, prime(2423 + 2) + 2 = 21599 + 2 = 21601 and 2*1727 - 2423 = 1031 all prime.

Mathematica

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n+2]+2]
f[n_, k_]:=2k+EulerPhi[n-k]-1
a[n_]:=Sum[If[p[f[n, k]]&&PrimeQ[2n-f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A001359, A002372, A002375, A006512, A234808, A236531, A236566, A236568, A236569.

Sequence in context: A194447 A307219 A345764 * A333252 A366376 A293375

Adjacent sequences: A236570 A236571 A236572 * A236574 A236575 A236576

Keyword

nonn

Author

Zhi-Wei Sun, Jan 29 2014

Status

approved