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A234360 a(n) = |{0 < k < n: (k+1)^{phi(n-k)} + 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 (list; graph; refs; listen; history; text; internal format)
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(n-k)/2} - k prime.

(ii) If n > 1, then k*(k+1)^{phi(n-k)} + 1 is prime for some 0 < k < n. If n > 3, then k*(k+1)^{phi(n-k)/2} - 1 is prime for some 0 < k < n.

LINKS

Zhi-Wei 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[n-k])+k

a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-1}]

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

Zhi-Wei Sun, Dec 24 2013

STATUS

approved

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Last modified August 22 11:34 EDT 2019. Contains 326176 sequences. (Running on oeis4.)