A236413 Positive integers m with p(m)^2 + q(m)^2 prime, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

1, 2, 3, 4, 6, 17, 24, 37, 44, 95, 121, 162, 165, 247, 263, 601, 714, 742, 762, 804, 1062, 1144, 1149, 1323, 1508, 1755, 1833, 1877, 2330, 2380, 2599, 3313, 3334, 3368, 3376, 3395, 3504, 3688, 3881, 4294, 4598, 4611, 5604, 5696, 5764, 5988, 6552, 7206, 7540, 7689

Offset: 1

Author

Zhi-Wei Sun, Jan 24 2014

Keywords

• nonn

Comments

According to the conjecture in A236412, this sequence should have infinitely many terms.

See A236414 for primes of the form p(m)^2 + q(m)^2.

See also A236440 for a similar sequence.

Examples

			a(1) = 1 since p(1)^2 + q(1)^2 = 1^2 + 1^2 = 2 is prime.
a(2) = 2 since p(2)^2 + q(2)^2 = 2^2 + 1^2 = 5 is prime.
a(3) = 3 since p(3)^2 + q(3)^2 = 3^2 + 2^2 = 13 is prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..200
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000009, A000010, A000040, A233346, A236412, A236414, A236417, A236418, A236419, A236440.

Programs

Mathematica

pq[n_]:=PrimeQ[PartitionsP[n]^2+PartitionsQ[n]^2]
n=0;Do[If[pq[m],n=n+1;Print[n," ",m]],{m,1,10000}]