

A234360


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


10



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

1,3


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 1. Also, for any n > 5 there is a positive integer k < n with (k+1)^{phi(nk)/2}  k prime.
(ii) If n > 1, then k*(k+1)^{phi(nk)} + 1 is prime for some 0 < k < n. If n > 3, then k*(k+1)^{phi(nk)/2}  1 is prime for some 0 < k < n.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..2500


EXAMPLE

a(74) = 2 since (2+1)^{phi(72)} + 2 = 3^{24} + 2 =
282429536483 and (14+1)^{phi(60)} + 14 = 15^{16} + 14 = 6568408355712890639 are both prime.


MATHEMATICA

f[n_, k_]:=f[n, k]=(k+1)^(EulerPhi[nk])+k
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n1}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000010, A000040, A234309, A234310, A234337, A234344, A234346, A234347, A234359
Sequence in context: A278288 A023154 A070820 * A317838 A213938 A031501
Adjacent sequences: A234357 A234358 A234359 * A234361 A234362 A234363


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 24 2013


STATUS

approved



