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A234347
a(n) = |{0 < k < n: 3^k + 3^{phi(n-k)/2} - 1 is prime}|, where phi(.) is Euler's totient function.
14
0, 0, 0, 1, 2, 3, 4, 3, 3, 5, 3, 5, 6, 7, 2, 6, 7, 11, 7, 3, 6, 8, 7, 4, 11, 8, 8, 6, 6, 10, 7, 6, 8, 5, 6, 4, 8, 4, 6, 6, 6, 11, 10, 3, 9, 6, 6, 4, 10, 6, 7, 3, 4, 9, 8, 9, 7, 9, 5, 9, 7, 9, 8, 4, 6, 9, 10, 7, 8, 9, 10, 5, 6, 12, 5, 6, 9, 10, 8, 9, 7, 8, 8, 10
OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 3.
See also the conjecture in A234337.
EXAMPLE
a(4) = 1 since 3^1 + 3^{phi(3)/2} - 1 = 5 is prime.
a(5) = 2 since 3^1 + 3^{phi(4)/2} - 1 = 5 and 3^2 + 3^{phi(3)/2} - 1 are both prime.
MATHEMATICA
f[n_, k_]:=3^k+3^(EulerPhi[n-k]/2)-1
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 24 2013
STATUS
approved