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A236389 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2^m*p(m) + 1 prime}|, where p(.) is the partition function (A000041). 2
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 2, 2, 1, 0, 0, 2, 1, 0, 4, 3, 2, 0, 2, 3, 2, 2, 4, 2, 2, 1, 3, 2, 1, 2, 3, 3, 5, 3, 3, 4, 2, 8, 3, 2, 4, 4, 2, 4, 3, 5, 3, 5, 5, 3, 7, 3, 6, 6, 6, 4, 4, 2, 9, 3, 5, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,38

COMMENTS

Conjecture: a(n) > 0 for all n > 56.

We have verified this for n up to 33000.

The conjecture implies that there are infinitely many positive integers m with 2^m*p(m) + 1 prime. See A236390 for a list of such numbers m.

LINKS

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

EXAMPLE

a(10) = 1 since phi(1)/2 + phi(9)/12 = 1/2 + 6/12 = 1 with 2^1*p(1) + 1 = 2 + 1 = 3 prime.

a(30) = 1 since phi(17)/2 + phi(13)/12 = 8 + 1 = 9 with 2^9*p(9) + 1 = 512*30 + 1 = 15361 prime.

a(8261) = 1 since phi(395)/2 + phi(8261-395)/12 = 156 + 198 = 354 with 2^(354)*p(354) + 1 = 2^(354)*363117512048110005 + 1 prime.

MATHEMATICA

q[n_]:=IntegerQ[n]&&PrimeQ[2^n*PartitionsP[n]+1]

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

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

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

CROSSREFS

Cf. A000010, A000040, A000041, A000079, A236390.

Sequence in context: A159459 A304095 A276675 * A143078 A106405 A228601

Adjacent sequences:  A236386 A236387 A236388 * A236390 A236391 A236392

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 24 2014

STATUS

approved

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Last modified December 6 14:15 EST 2019. Contains 329806 sequences. (Running on oeis4.)