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

2,2

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A000041, A259531, A261515.

Sequence in context: A022076 A055766 A267319 * A036204 A270814 A284597

Adjacent sequences: A261510 A261511 A261512 * A261514 A261515 A261516

Keyword

nonn

Author

Zhi-Wei Sun, Aug 22 2015

Status

approved