A259678 - OEIS

A259678

Primes of the form p(k)^2 + p(m)^2, where k and m are positive integers, and p(.) is the partition function given by A000041.

2, 5, 13, 29, 53, 229, 509, 709, 1021, 1789, 3137, 3257, 3361, 6829, 13337, 18229, 30977, 41177, 49201, 148229, 240101, 240109, 250301, 1004053, 1575029, 2511601, 3833989, 3851989, 6314389, 5934121, 9060109, 9148309, 13823549, 20842361, 31404937, 106714213, 116703973, 151536109, 221241901, 254416549

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OFFSET

1,1

COMMENTS

The conjecture in A259531 implies that the current sequence has infinitely many terms.

REFERENCES

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

LINKS

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

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

EXAMPLE

a(1) = 2 since p(1)^2 + p(1)^2 = 2 is prime.

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

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

a(4) = 29 since p(2)^2 + p(4)^2 = 2^2 + 5^2 = 29 is prime.

MATHEMATICA

n=0; Do[If[PrimeQ[PartitionsP[k]^2+PartitionsP[m]^2], n=n+1; Print[n, " ", PartitionsP[k]^2+PartitionsP[m]^2]], {m, 1, 34}, {k, 1, m}]

CROSSREFS

Cf. A000040, A000041, A259531.