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A236358 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2*3^m + 1 prime}|, where phi(.) is Euler's totient function. 4
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 1, 0, 2, 3, 1, 1, 0, 2, 1, 0, 2, 3, 3, 3, 2, 3, 2, 1, 4, 1, 4, 1, 4, 5, 3, 5, 7, 7, 8, 5, 5, 4, 4, 7, 7, 4, 7, 3, 6, 4, 5, 5, 6, 7, 6, 4, 5, 7, 6, 9, 5, 8, 7, 7, 4, 6, 5, 4, 6, 9, 8, 3, 6, 8, 9, 8, 8, 7, 8, 8, 9, 8, 4, 7, 4, 7, 7, 5, 4, 8, 6, 6, 7, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,16

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 26.

(ii) For any integer n > 37, there is a positive integer k < n such that m = phi(k)/2 + phi(n-k)/12 is an integer with 2*3^m - 1 prime.

We have verified both parts for n up to 50000. Clearly, part (i) implies that there are infinitely many positive integers m with 2*3^m + 1 prime, while part (ii) implies that there are infinitely many positive integers m with 2*3^m - 1 prime.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(36) = 1 since phi(15)/2 + phi(21)/12 = 4 + 1 = 5 with 2*3^5 + 1 = 487 prime.

MATHEMATICA

p[n_]:=IntegerQ[n]&&PrimeQ[2*3^n+1]

f[n_, k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12

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

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

CROSSREFS

Cf. A000010, A000040, A000244, A079363, A111974, A234451, A234503, A234504.

Sequence in context: A143067 A219605 A123949 * A144082 A145579 A167655

Adjacent sequences:  A236355 A236356 A236357 * A236359 A236360 A236361

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 23 2014

STATUS

approved

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Last modified August 22 06:01 EDT 2019. Contains 326172 sequences. (Running on oeis4.)