A236389 - OEIS

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).

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

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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 * A337821 A143078 A106405

Adjacent sequences: A236386 A236387 A236388 * A236390 A236391 A236392

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 24 2014

STATUS

approved