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%I #11 Apr 18 2019 14:42:03
%S 1,6,1,3,9,2,1,181,5,459,5,1,2,18,3,421,35,14,183,3274,12,143,501,422,
%T 1407,1,170,9,55,153,2044,426,274,74,17,7,68,452,1084,1637,3,6,43,
%U 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
%N Least positive integer k such that prime(k*n)-1 is a square, or 0 if no such k exists.
%C Conjecture: a(n) > 0 for all n > 0.
%C This is stronger than the well-known conjecture that there are infinitely many primes of the form x^2+1 with x an integer.
%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="/A259731/b259731.txt">Table of n, a(n) for n = 1..2000</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) = 1 since prime(1*1)-1 = 2-1 = 1^2.
%e a(2) = 6 since prime(6*2)-1 = 37-1 = 6^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t 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}]
%t lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[Prime[k*n]-1]],k++];k]; Array[ lpi,70] (* _Harvey P. Dale_, Apr 18 2019 *)
%Y Cf. A000040, A000290, A002496, A076948, A238573, A259712.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jul 04 2015
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