

A234470


Number of ways to write n = k + m with k > 0 and m > 2 such that p(k + phi(m)/2) is prime, where p(.) is the partition function (A000041) and phi(.) is Euler's totient function.


17



0, 0, 0, 1, 2, 3, 4, 5, 5, 4, 4, 4, 2, 2, 3, 5, 4, 2, 4, 2, 3, 2, 3, 2, 3, 1, 0, 3, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 4, 2, 1, 4, 2, 1, 2, 3, 3, 3, 1, 0, 4, 2, 4, 1, 1, 2, 2, 3, 2, 2, 0, 2, 2, 1, 2, 2, 1, 1, 2, 2, 4, 2, 1, 0, 1, 3, 1, 0, 2, 4, 3, 1, 6, 2, 2, 1, 2, 4, 3, 1, 2, 6, 2, 3, 2, 2, 2, 2, 3, 3
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OFFSET

1,5


COMMENTS

Conjecture: a(n) > 0 if n > 3 is not among 27, 34, 50, 61, 74, 78, 115, 120, 123, 127.
This implies that there are infinitely many primes in the range of the partition function p(n).


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..6500
Z.W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014


EXAMPLE

a(26) = 1 since 26 = 2 + 24 with p(2 + phi(24)/2) = p(6) = 11 prime.
a(54) = 1 since 54 = 27 + 27 with p(27 + phi(27)/2) = p(36) = 17977 prime.
a(73) = 1 since 73 = 1 + 72 with p(1 + phi(72)/2) = p(36) = 17977 prime.
a(110) = 1 since 110 = 65 + 45 with p(65 + phi(45)/2) = p(77) = 10619863 prime.
a(150) = 1 since 150 = 123 + 27 with p(123 + phi(27)/2) = p(132) = 6620830889 prime.
a(170) = 1 since 170 = 167 + 3 with p(167 + phi(3)/2) = p(168) = 228204732751 prime.


MATHEMATICA

f[n_, k_]:=PartitionsP[k+EulerPhi[nk]/2]
a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n3}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000010, A000040, A000041, A049575, A232504, A233307, A233346, A233918, A234200, A234246, A234309, A234337, A234344, A234347, A234359, A234360, A234451, A234475
Sequence in context: A301534 A101916 A100771 * A113771 A131845 A062186
Adjacent sequences: A234467 A234468 A234469 * A234471 A234472 A234473


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Dec 26 2013


STATUS

approved



