cp's OEIS Frontend

A261353 Least positive integer k such that prime(prime(k))*prime(prime(k*n)) = prime(p)+2 for some prime p.

%I A261353 #6 Aug 16 2015 12:07:01
%S A261353 11,2,1,606,350,166,53,1865,7,45,1308,68,215,61,256,13,248,90,1,1779,
%T A261353 796,1,4,444,650,55,157,303,82,84,25,3,1912,621,128,205,164,1091,61,
%U A261353 12,337,1,303,15,23,418,212,23,2494,1,472,771,1,36,8,46,8,18,264,22,725,85,65,231,606,3,1,43,144,164
%N A261353 Least positive integer k such that prime(prime(k))*prime(prime(k*n)) = prime(p)+2 for some prime p.
%C A261353 Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(prime(m))*prime(prime(n)) = prime(p)+2 for some prime p.
%C A261353 This implies that the sequence A261352 has infinitely many terms.
%D A261353 Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
%D A261353 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 A261353 Zhi-Wei Sun, <a href="/A261353/b261353.txt">Table of n, a(n) for n = 1..382</a>
%H A261353 Zhi-Wei Sun, <a href="/A261353/a261353.txt">Checking the conjecture for r = a/b with a,b = 1,...,31</a>
%H A261353 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 A261353 a(1) = 11 since prime(prime(11))*prime(prime(11*1)) = prime(31)^2 = 127^2 = 16129 = prime(1877)+2 with 1877 prime.
%e A261353 a(4) = 606 since prime(prime(606))*prime(prime(606*4)) = prime(4457)*prime(21589) = 42643*244471 = 10424976853 = prime(473490161)+2 with 473490161 prime.
%t A261353 f[n_]:=Prime[Prime[n]]
%t A261353 PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
%t A261353 Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n, " ", k];Continue,{n,1,70}]
%Y A261353 Cf. A000040, A109611, A261282, A261352.
%K A261353 nonn
%O A261353 1,1
%A A261353 _Zhi-Wei Sun_, Aug 15 2015