

A234344


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


14



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
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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

ZhiWei 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[nk]/2)
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n1}]
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 A279319
Adjacent sequences: A234341 A234342 A234343 * A234345 A234346 A234347


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 23 2013


STATUS

approved



