cp's OEIS Frontend

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

%I A261513 #15 Aug 23 2015 10:36:09
%S A261513 1,46,1,115,1,9,4,17,1,3,12,6,5,3,2,1253,1035,716,4028,6154,9,3,1219,
%T A261513 94,64,195,1545,9909,365,52,182,76,277,135,1321,1619,9693,5485,8001,
%U A261513 946,1,36,7154,10354,1,2157,33,1344,1,39,1698,732,24505,1,637,14,8,2127,1460
%N A261513 Least positive integer k with p(prime(k))+p(prime(k*n)) prime, where p(.) is the partition function given by A000041.
%C A261513 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.
%C A261513 This implies that there are infinitely many primes of the form p(q) + p(r) with q and r both prime.
%D A261513 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.
%H A261513 Zhi-Wei Sun, <a href="/A261513/b261513.txt">Table of n, a(n) for n = 2..100</a>
%H A261513 Zhi-Wei Sun, <a href="/A261513/a261513_1.txt">Checking the conjecture for r = a/b with 1 <= a < b <= 37</a>
%H A261513 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A261513 a(2) = 1 since p(prime(1)) + p(prime(1*2)) = p(2) + p(3) = 2 + 3 = 5 is prime.
%e A261513 a(3) = 46 since p(prime(46)) + p(prime(46*3)) = p(199) + p(787) = 3646072432125 + 3223934948277725160271634798 = 3223934948277728806344066923 is prime.
%t A261513 f[n_]:=PartitionsP[Prime[n]]
%t A261513 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}]
%Y A261513 Cf. A000040, A000041, A259531, A261515.
%K A261513 nonn
%O A261513 2,2
%A A261513 _Zhi-Wei Sun_, Aug 22 2015