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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 23 2013
STATUS
approved