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

Author

Bernard Schott, Sep 16 2021

Keywords

• nonn

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, ...

Examples

			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.
		

Links

• 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.

Crossrefs

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

Cf. A347191, A347193, A347194.

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

Programs

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 *)