A259731 Least positive integer k such that prime(k*n)-1 is a square, or 0 if no such k exists.

1, 6, 1, 3, 9, 2, 1, 181, 5, 459, 5, 1, 2, 18, 3, 421, 35, 14, 183, 3274, 12, 143, 501, 422, 1407, 1, 170, 9, 55, 153, 2044, 426, 274, 74, 17, 7, 68, 452, 1084, 1637, 3, 6, 43, 1141, 1, 8218, 1860, 211, 42, 1582, 53, 813, 2, 85, 1, 5714, 61, 1379, 296, 1457, 57, 1022, 4, 213, 1331, 137, 525, 37, 167, 1130

Offset: 1

Author

Zhi-Wei Sun, Jul 04 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0.

This is stronger than the well-known conjecture that there are infinitely many primes of the form x^2+1 with x an integer.

Examples

			a(1) = 1 since prime(1*1)-1 = 2-1 = 1^2.
a(2) = 6 since prime(6*2)-1 = 37-1 = 6^2.
		

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..2000
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A002496, A076948, A238573, A259712.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=0;Label[bb];k=k+1;If[SQ[Prime[k*n]-1],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k];Continue,{n,1,70}]
lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k*n]-1]],k++];k]; Array[ lpi,70] (* Harvey P. Dale, Apr 18 2019 *)