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%I #41 Jul 15 2015 04:50:40
%S 6,3,8,71,12,14,105,221,24,499,261,612,1341,175,917,549,1351,2303,
%T 2273,4767,364,1395,1390,1431,6481,2479,918,2412,17783,3178,2994,7538,
%U 3409,1361,9645,3454,9197,7074,10418,6059,36235,182,1910,4648,1130,695,3973,10839,8647,7942
%N Least positive integer k such that prime(k*n) - 1 = (prime(i*n)-1)*(prime(j*n)-1) for some integers 0 < i < j < k.
%C Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n, the set {prime(k*n)+m: k = 1,2,3,...} always contains three distinct elements x, y and z with x*y = z.
%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="/A257938/b257938.txt">Table of n, a(n) for n = 1..200</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(1) = 6 since prime(6*1)-1 = 12 = 2*6 = (prime (2*1)-1)*(prime(4*1)-1).
%e a(4) = 71 since prime(71*4)-1 = 1860 = 6*310 = (prime(1*4)-1)*(prime(16*4)-1).
%e a(41) = 36235 since prime(36235*41)-1 = 23634312 = 676*34962 = (prime(3*41)-1)*(prime(91*41)-1).
%e a(69) = 64999 since prime(64999*69)-1 = 76643820 = 4590*16698 = (prime(9*69)-1)*(prime(28*69)-1).
%e a(77) = 137789 since prime(137789*77)-1 = 191037600 = 2028*94200 = (prime(4*77)-1)*(prime(118*77)-1).
%e a(99) = 167708 since prime(167708*99)-1 = 306849088 = 10528*29146 = (prime(13*99)-1)*(prime(32*99)-1).
%e a(189) = 951492 since prime(951492*189)-1 = 3776304996 = 4126*915246 = (prime(3*189)-1)*(prime(383*189)-1).
%t Dv[n_]:=Divisors[Prime[n]-1]
%t L[n_]:=Length[Dv[n]]
%t P[k_,n_,i_]:=PrimeQ[Part[Dv[k*n],i]+1]&&Mod[PrimePi[Part[Dv[k*n],i]+1],n]==0
%t Do[k=0;Label[bb];k=k+1; Do[If[P[k,n,i]&&P[k,n,L[k*n]-i+1],Goto[aa]],{i,2,L[k*n]/2}];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,50}]
%o (PARI) a(n)={my(i,j,k=3);while(1,for(j=2,k-1,for(i=1,j-1,if(prime(k*n) - 1 == (prime(i*n)-1)*(prime(j*n)-1),break(3));));k++);return(k);}
%o main(size)={return(vector(size,n,a(n)));} /* _Anders Hellström_, Jul 13 2015 */
%Y Cf. A000040, A257928.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jul 13 2015
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