A257938 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.

6, 3, 8, 71, 12, 14, 105, 221, 24, 499, 261, 612, 1341, 175, 917, 549, 1351, 2303, 2273, 4767, 364, 1395, 1390, 1431, 6481, 2479, 918, 2412, 17783, 3178, 2994, 7538, 3409, 1361, 9645, 3454, 9197, 7074, 10418, 6059, 36235, 182, 1910, 4648, 1130, 695, 3973, 10839, 8647, 7942

Offset: 1

Author

Zhi-Wei Sun, Jul 13 2015

Keywords

• nonn

Comments

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.

Examples

			a(1) = 6 since prime(6*1)-1 = 12 = 2*6 = (prime (2*1)-1)*(prime(4*1)-1).
a(4) = 71 since prime(71*4)-1 = 1860 = 6*310 = (prime(1*4)-1)*(prime(16*4)-1).
a(41) = 36235 since prime(36235*41)-1 = 23634312 = 676*34962 = (prime(3*41)-1)*(prime(91*41)-1).
a(69) = 64999 since prime(64999*69)-1 = 76643820 = 4590*16698 = (prime(9*69)-1)*(prime(28*69)-1).
a(77) = 137789 since prime(137789*77)-1 = 191037600 = 2028*94200 = (prime(4*77)-1)*(prime(118*77)-1).
a(99) = 167708 since prime(167708*99)-1 = 306849088 = 10528*29146 = (prime(13*99)-1)*(prime(32*99)-1).
a(189) = 951492 since prime(951492*189)-1 = 3776304996 = 4126*915246 = (prime(3*189)-1)*(prime(383*189)-1).
		

References

• 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.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..200
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A257928.

Programs

Mathematica

Dv[n_]:=Divisors[Prime[n]-1]
L[n_]:=Length[Dv[n]]
P[k_,n_,i_]:=PrimeQ[Part[Dv[k*n],i]+1]&&Mod[PrimePi[Part[Dv[k*n],i]+1],n]==0
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}]

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);}
main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 13 2015 */