A234344 a(n) = |{0 < k < n: 2^{phi(k)/2} + 3^{phi(n-k)/2} is prime}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 6, 8, 7, 9, 12, 12, 10, 10, 10, 10, 16, 7, 11, 9, 6, 14, 11, 17, 12, 15, 15, 17, 16, 15, 19, 18, 12, 13, 9, 20, 11, 8, 17, 19, 19, 12, 17, 14, 16, 9, 21, 16, 13, 12, 16, 19, 17, 11, 21, 15, 16, 15, 17, 19, 16, 23, 11, 20, 15

Offset

1,7

Comments

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

This implies that there are infinitely many primes of the form 2^k + 3^m, where k and m are positive integers.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

Example

a(6) = 1 since 2^{phi(3)/2} + 3^{phi(3)/2} = 5 is prime.
a(8) = 3 since 2^{phi(3)/2} + 3^{phi(5)/2} = 11, 2^{phi(4)/2} + 3^{phi(4)/2} = 5, and 2^{phi(5)/2} + 3^{phi(3)/2} = 7 are all prime.

Mathematica

f[n_, k_]:=2^(EulerPhi[k]/2)+3^(EulerPhi[n-k]/2)
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A000079, A000244, A004051, A234309, A234310, A234337, A234346, A234347

Sequence in context: A097377 A234741 A063917 * A331298 A325351 A363776

Adjacent sequences: A234341 A234342 A234343 * A234345 A234346 A234347

Keyword

nonn

Author

Zhi-Wei Sun, Dec 23 2013

Status

approved