A237127 Number of ways to write n = k + m (0 < k < m) with k and m terms of A072281.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 3, 2, 3, 3, 3, 2, 2, 4, 3, 3, 2, 2, 4, 3, 4, 3, 4, 4, 3, 3, 4, 5, 4, 1, 3, 3, 5, 4, 4, 4, 4, 5, 3, 4, 2, 4, 4, 4, 5, 2, 4, 1, 4, 4, 4, 4, 1, 3, 4, 4, 5, 5

Offset

1,15

Comments

Conjecture: a(n) > 0 for all n > 11.

Clearly, this implies the twin prime conjecture.

Links

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

Example

 a(13) = 1 since 13 = 5 + 8 with phi(5) - 1 = 3, phi(5) + 1 = 5, phi(8) - 1 = 3 and phi(8) + 1 = 5 all prime.
a(60) = 1 since 60 = 18 + 42 with phi(18) - 1 = 5, phi(18) + 1 = 7, phi(42) - 1 = 11 and phi(42) + 1 = 13 all prime.
a(84) = 1 since 84 = 7 + 77 with phi(7) - 1 = 5, phi(7) + 1 = 7, phi(77) - 1 = 59 and phi(77) + 1 = 61 all prime.

Mathematica

PQ[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[PQ[k]&&PQ[n-k], 1, 0], {k, 1, (n-1)/2}]
Table[a[n], {n, 1, 70}]

Crossrefs

Cf. A000010, A000040, A001359, A006512, A014574, A072281.

Sequence in context: A110917 A329340 A070956 * A262746 A007828 A378117

Adjacent sequences: A237124 A237125 A237126 * A237128 A237129 A237130

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2014

Status

approved