A257856 Least positive integer k such that prime(k*n) - prime(k) is a square.

1, 1, 5, 2, 1, 291, 4, 31, 4, 131, 66, 8, 113, 4, 1770, 19, 122, 27, 509, 61, 138, 1484, 1, 508, 118, 1033, 48, 314, 78, 1522, 4, 8, 169, 341, 650, 37, 3456, 1172, 221, 21, 119, 105, 34, 670, 196, 19, 30, 4, 1, 88, 496, 30, 1460, 90, 12, 1270, 812, 2096, 311, 131, 95, 241, 198, 34, 19, 63, 8, 75, 2, 10413

Offset: 1

Author

Zhi-Wei Sun, Jul 12 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. In general, every rational number r > 1 can be written as m/n with m > n > 0 such that prime(m) - prime(n) is a square.

This conjecture is a supplement to the conjecture in A259712.

Examples

			a(3) = 5 since prime(5*3) - prime(5) = 47 - 11 = 6^2.
a(70) = 10413 since prime(10413*70) - prime(10413) = 11039173 - 109537 = 3306^2.
a(1133) = 697092 since prime(697092*1133) - prime(697092) = 17813555143 - 10523959 = 133428^2.
		

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..1250
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A257663, A259712.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=0;Label[bb];k=k+1;If[SQ[Prime[n*k]-Prime[k]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]
lpi[n_]:=Module[{k=1,sq},sq=Prime[k*n]-Prime[k];While[!IntegerQ[ Sqrt[ sq]], k++;sq=Prime[k*n]-Prime[k]];k]; Array[lpi, 70] (* Harvey P. Dale, Oct 15 2015 *)