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%I #10 Aug 15 2015 14:03:23
%S 1,1,47500,20440,2,124560,17850,2730,185550,1,518910,429180,10,687480,
%T 81030,36,1568340,2,1165750,7410,10,6780,481140,10,10,5430,240,2730,
%U 72660,2080,18700,291720,295080,52860,5430,1,81030,56400,12490,43590,124560,40030,5170,278700,2091850,131320,184110,11206510,12910,1245780
%N Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.
%C Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n with m and n in the set {k>0: k+1, k^2+1 and k^2+prime(k)^2 are all prime}.
%C For example, 5/8 = 3567600/5708160 with 3567600+1, 3567600^2+1 = 12727769760001, 3567600^2 + prime(3567600)^2 = 3567600^2 + 60098671^2 = 3624578025726241, 5708160+1, 5708160^2+1 = 32583090585601, and 5708160^2 + prime(5708160)^2 = 5708160^2 + 99018553^2 = 9837256928799409 all prime.
%C The conjecture implies that there are infinitely many primes p with (p-1)^2+1 and (p-1)^2+prime(p-1)^2 both prime.
%C We also guess that any positive rational number can be written as m/n, where m and n are positive integers with m^2+prime(m)^2, m^2+prime(n)^2, n^2+prime(m)^2 and n^2+prime(n)^2 all prime.
%D 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 Zhi-Wei Sun, <a href="/A261339/b261339.txt">Table of n, a(n) for n = 1..1000</a>
%H Zhi-Wei Sun, <a href="/A261339/a261339.txt">Checking the conjecture for r = a/b with a,b = 1..60</a>
%H 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 a(3) = 47500 since 47501, 47500^2 + 1 = 2256250001, 47500^2 + prime(47500)^2 = 47500^2 + 578827^2 = 337296945929, 47500*3 + 1 = 142501, (47500*3)^2 + 1 = 20306250001, and (47500*3)^2 + prime(47500*3)^2 = 142500^2 + 1907023^2 = 3657042972529 are all prime.
%t PQ[n_]:=PrimeQ[n+1]&&PrimeQ[n^2+1]&&PrimeQ[n^2+Prime[n]^2]
%t Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,50}]
%Y Cf. A000040, A005574, A236193, A259492, A259540, A260120.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Aug 15 2015