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, 5934121, 6314389, 9060109, 9148309, 13823549, 20842361, 31404937, 106714213, 116703973, 151536109, 221241901, 221743789, 230563789, 254416549, 323412629, 323799793, 332232629, 336996053, 468159773, 676780229, 676783361, 682714321

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

p[n_]:=p[n]=PartitionsP[n];
tab={}; Do[s=p[m]^2+p[k]^2; If[PrimeQ[s], tab=Append[tab, s]], {m, 1, 1600}, {k, 1, m}];
LL:=LL=Sort[DeleteDuplicates[tab]]; Do[Print[n, LL[[n]]], {n, 1, 10000}]
Take[Select[Total/@Tuples[PartitionsP[Range[2000]]^2, 2], PrimeQ]//Union, 50] (* Harvey P. Dale, Jan 24 2025 *)

Crossrefs

Cf. A000040, A000041, A259531.

Sequence in context: A093702 A011756 A050950 * A218152 A079991 A393445

Adjacent sequences: A259675 A259676 A259677 * A259679 A259680 A259681

Keyword

nonn

Author

Zhi-Wei Sun, Jul 03 2015

Extensions

Corrected and extended by Harvey P. Dale, Jan 24 2025

Status

approved