A347192 Integers k such that the number of divisors of k^2 - 1 (A347191) sets a new record.

2, 3, 5, 7, 11, 17, 19, 29, 41, 71, 109, 161, 169, 181, 379, 449, 649, 701, 881, 1079, 1189, 1871, 2449, 3079, 4159, 5851, 11969, 19601, 23561, 23869, 24751, 43471, 82081, 94249, 157249, 222641, 252449, 313039, 627199, 677249, 790399, 1276001, 2308879, 4058209

Offset

1,1

Comments

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

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

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

Links

Table of n, a(n) for n=1..44.

Diophante, A1885, Cachés derrière leurs diviseurs (in French).

Adrian Dudek, On the Number of Divisors of n^2-1, arXiv:1507.08893 [math.NT], 2015.

Example

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.

Mathematica

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

Crossrefs

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

Cf. A347191, A347193, A347194.

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

Sequence in context: A262835 A258261 A228424 * A335325 A189828 A090481

Adjacent sequences: A347189 A347190 A347191 * A347193 A347194 A347195

Keyword

nonn

Author

Bernard Schott, Sep 16 2021

Status

approved