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A261513 Least positive integer k with p(prime(k))+p(prime(k*n)) prime, where p(.) is the partition function given by A000041. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 2 08:19 EST 2021. Contains 349437 sequences. (Running on oeis4.)