A261513 Least positive integer k with p(prime(k))+p(prime(k*n)) prime, where p(.) is the partition function given by A000041.

1, 46, 1, 115, 1, 9, 4, 17, 1, 3, 12, 6, 5, 3, 2, 1253, 1035, 716, 4028, 6154, 9, 3, 1219, 94, 64, 195, 1545, 9909, 365, 52, 182, 76, 277, 135, 1321, 1619, 9693, 5485, 8001, 946, 1, 36, 7154, 10354, 1, 2157, 33, 1344, 1, 39, 1698, 732, 24505, 1, 637, 14, 8, 2127, 1460

Offset: 2

Author

Zhi-Wei Sun, Aug 22 2015

Keywords

• nonn

Comments

Conjecture: Any positive rational number r not equal to one can be written as m/n, where m and n are positive integers with p(prime(m)) + p(prime(n)) prime.

This implies that there are infinitely many primes of the form p(q) + p(r) with q and r both prime.

Examples

			a(2) = 1 since p(prime(1)) + p(prime(1*2)) = p(2) + p(3) = 2 + 3 = 5 is prime.
a(3) = 46 since p(prime(46)) + p(prime(46*3)) = p(199) + p(787) = 3646072432125 + 3223934948277725160271634798 = 3223934948277728806344066923 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 = 2..100
• Zhi-Wei Sun, Checking the conjecture for r = a/b with 1 <= a < b <= 37
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000041, A259531, A261515.

Programs

Mathematica

f[n_]:=PartitionsP[Prime[n]]
Do[k=0;Label[bb];k=k+1;If[PrimeQ[f[k]+f[k*n]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,2,60}]