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
Keywords
Examples
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.
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.
Programs
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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 *)
Extensions
Corrected and extended by Harvey P. Dale, Jan 24 2025
Comments