A189828 -id:A189828 - cp's OEIS frontend

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

Bernard Schott, Sep 16 2021

nonn

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, ...

			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.

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.

Cf. A000005, A005563, A014574, A129296, A129297.
Cf. A347191, A347193, A347194.
Cf. A090481, A189828, A335325 (similar, with k = p prime).

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 *)

A189829 Record values of d(n-1) + d(n+1), where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 12, 14, 16, 20, 22, 24, 26, 28, 32, 34, 36, 38, 44, 46, 48, 52, 56, 58, 60, 68, 72, 76, 78, 80, 84, 88, 92, 96, 98, 100, 104, 106, 108, 116, 124, 128, 136, 148, 156, 160, 168, 172, 174, 176, 184, 196, 200, 208, 224, 236, 248, 260, 268

Offset: 1

T. D. Noe, Apr 28 2011

nonn

The n are in A189828.

Donovan Johnson, Table of n, a(n) for n = 1..135

Cf. A189827, A189828.

cp's OEIS Frontend

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

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