A347192 - OEIS

%I #22 Dec 04 2023 14:29:29

%S 2,3,5,7,11,17,19,29,41,71,109,161,169,181,379,449,649,701,881,1079,

%T 1189,1871,2449,3079,4159,5851,11969,19601,23561,23869,24751,43471,

%U 82081,94249,157249,222641,252449,313039,627199,677249,790399,1276001,2308879,4058209

%N Integers k such that the number of divisors of k^2 - 1 (A347191) sets a new record.

%C The first ten terms are the same as A090481 and A189828, then a(11) = 109 while A090481(11) = 179 and A189828(11) = 161.

%C The first eleven terms are the same as A335325, then a(12) = 161, which is nonprime, while A335325(12) = 181.

%C The corresponding records obtained are 2, 4, 8, 10, 16, 18, 24, 32, 40, 60, 64, 70, 80, 96, ...

%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/3848-a1885-caches-derriere-leurs-diviseurs">A1885, Cachés derrière leurs diviseurs</a> (in French).

%H Adrian Dudek, <a href="http://arxiv.org/abs/1507.08893">On the Number of Divisors of n^2-1</a>, arXiv:1507.08893 [math.NT], 2015.

%e tau(71^2-1) = 60 and there is no integer k < 71 such that tau(k^2-1) >= 60, hence 71 is a term and a(10) = 71.

%t s[n_] := DivisorSigma[0, n^2 - 1]; sm = 0; seq = {}; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 2, 10^6}]; seq (* _Amiram Eldar_, Sep 16 2021 *)

%t DeleteDuplicates[Table[{k,DivisorSigma[0,k^2-1]},{k,2,4060000}],GreaterEqual[#1[[2]],#2[[2]]]&] [[;;,1]] (* _Harvey P. Dale_, Dec 04 2023 *)

%Y Cf. A000005, A005563, A014574, A129296, A129297.

%Y Cf. A347191, A347193, A347194.

%Y Cf. A090481, A189828, A335325 (similar, with k = p prime).

%K nonn

%O 1,1

%A _Bernard Schott_, Sep 16 2021