

A234337


a(n) = {0 < k < n  2: 4^k + 2^{phi(nk)}  1 is prime}, where phi(.) is Euler's totient function.


14



0, 0, 0, 1, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 8, 7, 6, 8, 6, 10, 8, 5, 6, 7, 10, 7, 6, 10, 9, 6, 7, 8, 12, 5, 9, 4, 9, 4, 6, 3, 8, 8, 11, 10, 9, 7, 7, 13, 12, 6, 7, 8, 6, 6, 13, 10, 8, 9, 9, 12, 6, 11, 14, 9, 5, 11, 7, 7, 10, 11, 7, 9, 10, 5, 9, 8, 8, 13, 7, 13
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OFFSET

1,5


COMMENTS

Conjecture: Let a be 2 or 3 or 4. If n > 3, then a^k + a^{phi(nk)/2}  1 is prime for some 0 < k < n  2.
This conjecture for a = 4 implies that there are infinitely many terms of the sequence A234310. The conjecture for a = 3 implies that there are infinitely many primes of the form 3^k + 3^m  1 (cf. A234346), where k and m are positive integers.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..4000


EXAMPLE

a(4) = 1 since 4^1 + 2^{phi(3)}  1 = 7 is prime.
a(5) = 2 since 4^1 + 2^{phi(4)}  1 = 7 and 4^2 + 2^{phi(3)}  1 = 19 are both prime.


MATHEMATICA

f[n_, k_]:=4^k+2^(EulerPhi[nk])1
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n3}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000010, A000040, A000244, A000302, A234309, A234310, A234344, A234346, A234347, A234359, A234360, A234361
Sequence in context: A227559 A234968 A141784 * A216391 A014202 A309247
Adjacent sequences: A234334 A234335 A234336 * A234338 A234339 A234340


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 23 2013


STATUS

approved



