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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
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
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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 03 2015
STATUS
approved