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A234359 a(n) = |{2 < k < n-2: 5^{phi(k)} + 5^{phi(n-k)/2} - 1 is prime}|, where phi(.) is Euler's totient function. 10
0, 0, 0, 0, 0, 1, 2, 1, 2, 4, 2, 4, 4, 3, 4, 3, 6, 5, 4, 6, 7, 8, 6, 7, 11, 7, 10, 9, 9, 7, 10, 11, 8, 7, 11, 10, 9, 6, 11, 15, 4, 14, 5, 14, 11, 13, 9, 13, 6, 12, 10, 12, 11, 10, 10, 13, 9, 7, 11, 7, 11, 4, 11, 9, 10, 6, 11, 8, 4, 10, 12, 13, 9, 7, 9, 6, 12, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: For any integer a > 1, there is a positive integer N(a) such that if n > N(a) then a^{phi(k)} + a^{phi(n-k)/2} - 1 is prime for some 2 < k < n-2. Moreover, we may take N(2) = N(3) = ... = N(6) = N(8) = 5 and N(7) = 17.

Clearly, this conjecture implies that for each  a = 2, 3, ... there are infinitely many primes of the form a^{2*k} + a^m - 1, where k and m are positive integers.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500

EXAMPLE

a(6) = 1 since 5^{phi(3)} + 5^{phi(3)/2} - 1 = 29 is prime.

a(11) = 2 since 5^{phi(4)} + 5^{phi(7)/2} - 1 = 149 and 5^{phi(7)} + 5^{phi(4)/2} - 1 = 15629 are both prime.

MATHEMATICA

f[n_, k_]:=5^(EulerPhi[k])+5^(EulerPhi[n-k]/2)-1

a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 3, n-3}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000010, A000040, A000351, A234309, A234310, A234337, A234344, A234346, A234347

Sequence in context: A121339 A307236 A324382 * A099500 A120253 A307018

Adjacent sequences:  A234356 A234357 A234358 * A234360 A234361 A234362

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Dec 24 2013

STATUS

approved

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Last modified August 20 20:50 EDT 2019. Contains 326155 sequences. (Running on oeis4.)