A237531 a(n) = |{0 < k < n/2: phi(k*(n-k)) - 1 and phi(k*(n-k)) + 1 are both prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 3, 1, 3, 1, 4, 1, 1, 1, 1, 4, 3, 2, 1, 2, 1, 1, 3, 2, 5, 4, 3, 3, 6, 2, 3, 1, 5, 4, 5, 2, 3, 5, 5, 3, 7, 6, 3, 7, 3, 8, 7, 4, 4, 5, 6, 4, 3, 9, 6, 8, 9, 8, 9, 9, 10, 7, 6, 3, 5, 4, 8, 4, 8, 5, 10, 2, 7, 9, 5, 7

Offset

1,6

Comments

Conjecture: a(n) > 0 for all n > 5.

Clearly, this implies the twin prime conjecture.

Links

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(12) = 1 since 12 = 3 + 9 with phi(3*9) - 1 = 17 and phi(3*9) + 1 = 19 both prime.
a(19) = 1 since 19 = 1 + 18 with phi(1*18) - 1 = 5 and phi(1*18) + 1 = 7 both prime.
a(86) = 1 since 86 = 8 + 78 with phi(8*78) - 1 = 191 and phi(8*78) + 1 = 193 both prime.

Mathematica

p[n_]:=PrimeQ[EulerPhi[n]-1]&&PrimeQ[EulerPhi[n]+1]
a[n_]:=Sum[If[p[k(n-k)], 1, 0], {k, 1, (n-1)/2}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000010, A000040, A001359, A006512, A014574, A072281, A233547, A234200, A237127, A237130, A237168, A237523.

Sequence in context: A050677 A058013 A223934 * A238504 A031356 A304522

Adjacent sequences: A237528 A237529 A237530 * A237532 A237533 A237534

Keyword

nonn

Author

Zhi-Wei Sun, Feb 09 2014

Status

approved