A259531 Least positive integer k such that p(k)^2 + p(k*n)^2 is prime, where p(.) is the partition function given by A000041, or 0 if no such k exists.

1, 1, 14, 11, 6, 31, 2, 34, 2, 76, 1, 100, 71, 38, 1, 7, 62, 1128, 1, 180, 123, 15, 174, 128, 4, 111, 110, 2, 4, 2, 2241, 21, 144, 416, 397, 31, 11, 8, 15, 5, 91, 56, 53, 23, 89, 18, 25, 341, 12, 1, 66, 454, 159, 36, 573, 26, 2, 488, 72, 416, 802, 440, 28, 30, 595, 17, 236, 947, 1289, 1287, 1000, 367, 80, 407, 1, 77, 938, 150, 36, 1

Offset: 1

Author

Zhi-Wei Sun, Jul 02 2015

Keywords

• nonn

Comments

Conjecture: Any positive rational number r can be written as m/n, where m and n are positive integers with p(m)^2 + p(n)^2 prime.

For example, 4/5 = 124/155, and the number p(124)^2 + p(155)^2 = 2841940500^2 + 66493182097^2 = 4429419891190341567409 is prime.

We also guess that any positive rational number can be written as m/n, where m and n are positive integers with p(m)+p(n) (or p(m)*p(n)-1, or p(m)*p(n)+1) prime.

Examples

			a(5) = 6 since p(6)^2 + p(6*5)^2 = 11^2 + 5604^2 = 31404937 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..200
• Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..100
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000041, A258803, A258836, A259487, A259492, A259540.

Programs

Mathematica

Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[k]^2+PartitionsP[k*n]^2],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k]; Continue,{n,1,80}]