A236412 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/8 is an integer with p(m)^2 + q(m)^2 prime}|, where phi(.) is Euler's totient, p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 0, 1, 2, 3, 3, 4, 5, 3, 4, 4, 7, 4, 5, 5, 3, 3, 4, 5, 4, 3, 6, 8, 3, 3, 3, 7, 3, 7, 4, 5, 3, 6, 3, 2, 3, 6, 3, 3, 2, 5, 1, 4, 6, 4, 3, 3, 7, 5, 3, 3, 3, 4, 1, 5, 4, 3, 2, 4, 3, 6, 2, 5, 6, 4, 5, 2, 1, 6, 4, 4, 2, 11, 1, 6, 3, 5, 6, 7, 2, 4, 4, 2, 3, 2

Offset

1,14

Comments

Conjecture: a(n) > 0 for all n > 17.

We have verified this for n up to 65000.

The conjecture implies that there are infinitely positive integers m with p(m)^2 + q(m)^2 prime. See A236413 for a list of such numbers m. See also A236414 for primes of the form p(m)^2 + q(m)^2.

Links

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

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Example

a(15) = 1 since phi(2)/2 + phi(13)/8 = 1/2 + 12/8 = 2 with p(2)^2 + q(2)^2 = 2^2 + 1^2 = 5 prime.
a(69) = 1 since phi(5)/2 + phi(64)/8 = 2 + 4 = 6 with p(6)^2 + q(6)^2 = 11^2 + 4^2 = 137 prime.
a(89) = 1 since phi(73)/2 + phi(16)/8 = 36 + 1 = 37 with p(37)^2 + q(37)^2 = 21637^2 + 760^2 = 468737369 prime.

Mathematica

p[n_]:=IntegerQ[n]&&PrimeQ[PartitionsP[n]^2+PartitionsQ[n]^2]
f[n_, k_]:=EulerPhi[k]/2+EulerPhi[n-k]/8
a[n_]:=Sum[If[p[f[n, k]], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000010, A000040, A000041, A233307, A236374, A236413, A236414.

Sequence in context: A303979 A301573 A061670 * A236263 A337126 A319572

Adjacent sequences: A236409 A236410 A236411 * A236413 A236414 A236415

Keyword

nonn

Author

Zhi-Wei Sun, Jan 24 2014

Status

approved