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%I #13 Jul 17 2015 04:44:28
%S 1,2,14,1,12,9,30,198,69,83,66,132,44,15,4,99,71,88,339,230,10,33,167,
%T 66,42,22,126,442,318,1185,29,289,37,174,157,44,146,301,171,403,2,5,
%U 26,699,573,144,338,33,2032,1212,404,11,135,267,380,221,447,159,898,1397
%N Least integer k > 0 such that (prime(k*n)-1)^2 = prime(j*n)-1 for some j > 0.
%C Conjecture: a(n) exists for any n > 0. In general, if a,b,c and m are integers with a > 0, gcd(a,b,c-m) = 1 and c == (a+b+1)*(m+1) (mod 2) such that b^2-4a*(c-m) is not a square and gcd(a*m-b,b^2+b-a*c-1) is not divisible by 3, then for any positive integer n there are two elements x and y of the set {prime(k*n)+m: k = 1,2,3,...} with a*x^2+b*x+c = y.
%C This implies the conjecture in A259731.
%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="/A260120/b260120.txt">Table of n, a(n) for n = 1..100</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) = 14 since (prime(14*3)-1)^2 = 180^2 = prime(3477)-1 = prime(1159*3)-1.
%e a(63) = 5162 since (prime(5162*63)-1)^2 = 4642456^2 = 21552397711936 = prime(726521033763)-1 = prime(11532079901*63)-1.
%t P[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
%t Do[k=0;Label[aa];k=k+1; If[P[n,(Prime[k*n]-1)^2+1],Goto[bb]];Goto[aa];Label[bb];Print[n, " ", k];Continue,{n,1,60}]
%Y Cf. A000040, A002496, A259731, A260082, A260121.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jul 17 2015