A347194 -id:A347194 - cp's OEIS frontend

A347191 Number of divisors of n^2-1.

2, 4, 4, 8, 4, 10, 6, 10, 6, 16, 4, 16, 8, 12, 8, 18, 4, 24, 8, 16, 8, 20, 6, 20, 12, 16, 8, 32, 4, 28, 8, 14, 16, 24, 8, 24, 8, 20, 8, 40, 4, 32, 12, 16, 12, 24, 6, 36, 12, 24, 8, 32, 8, 40, 16, 20, 8, 32, 4, 32, 12, 16, 24, 32, 8, 32, 8, 32, 8, 60, 4, 30, 12, 16, 24, 32, 8, 48, 10, 24

Offset: 2

Bernard Schott, Aug 22 2021

nonn

Inspired by problem A1885 in Diophante (see link).
As n^2-1 > 0 is never square, all terms are even.
a(n) = 2 iff n = 2.
a(n) = 4 iff n = 3 or iff n is average of twin prime pairs 'n-1' and 'n+1'; i.e. n is a member of ({3} Union A014574) or equivalently n is a term of A129297 \ {0,1,2}.
a(n) = 6 iff n is such that the two adjacent integers of n are a prime and a square of another prime: 8, 10, 24, 48, 168, 360, ... (A347194).

			a(5) = tau(5^2-1) = tau(24) = 8.
a(18) = tau(18^2-1) = tau(17*19) = 4, 18 is average of twin primes 17 and 19.

Amiram Eldar, Table of n, a(n) for n = 2..10000
Diophante, A1885, Cachés derrière leurs diviseurs (in French).
Adrian W. Dudek, On the number of divisors of n^2-1, Bulletin of the Australian Mathematical Society, Vol. 93, No. 2 (2016), pp. 194-198; arXiv preprint, arXiv:1507.08893 [math.NT], 2015.

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

Cf. A347192 (records), A347193 (smallest k with a(k) = n), A347194 (a(n)=6).

with(numtheory):
seq(tau(n^2-1), n=2..81);

a[n_] := Length[Divisors[n^2 - 1]]; Table[a[n], {n, 2, 81}] (* Robert P. P. McKone, Aug 22 2021 *)
Table[DivisorSigma[0, n^2 - 1], {n, 2, 100}] (* Vaclav Kotesovec, Aug 23 2021 *)

a(n) = numdiv(n^2-1); \\ Michel Marcus, Aug 23 2021

a(n)=my(a=valuation(n-1,2),b=valuation(n+1,2)); numdiv((n-1)>>a)*numdiv((n+1)>>b)*(a+b+1) \\ Charles R Greathouse IV, Sep 17 2021

first(n)=my(v=vector(n-1),x=[1,factor(1)],y=[2,factor(2)]); forfactored(k=3,n+1,  my(e=max(valuation(x[1],2), valuation(k[1],2))); v[k[1]-2]=numdiv(k)*numdiv(x)*(e+2)/(2*e+2); x=y; y=k); v \\ Charles R Greathouse IV, Sep 17 2021

from math import prod
from sympy import factorint
def a(n):
    ft = factorint(n+1, multiple=True) + factorint(n-1, multiple=True)
    return prod((e + 1) for e in (ft.count(f) for f in set(ft)))
print([a(n) for n in range(2, 82)]) # Michael S. Branicky, Sep 17 2021

a(n) = A000005(A005563(n-1)).
a(n) = 2 * A129296(n).
Sum_{k=2..n} a(k) ~ (6/Pi^2) * n*log(n)^2 (Dudek, 2016). - Amiram Eldar, Apr 07 2023

A347193 a(n) is the smallest m such that A347191(m) = 2*n, where A347191(m) = tau(m^2 - 1).

Original entry on oeis.org

2, 3, 8, 5, 7, 15, 33, 11, 17, 23, 513, 19, 13841287200, 31, 73, 29, 650377879817809571042122834560, 49, 131073, 41, 97, 1537, 31381059608, 79, 50626, 10239, 127, 223, 459986536544739960976800, 71, 8193465725814765556554001028792218848, 109, 61953, 163839, 161

Offset: 1

Bernard Schott, Sep 17 2021

nonn

Generalization of the questions proposed in Diophante problem A1885 (see links).
When p is prime, as a(p) is the smallest m such that tau(m^2 - 1) = 2*p, hence m^2 - 1 is of the form q * r^(p-1) with q > r primes, so we must solve the Diophantine equation (m-1)*(m+1) = q * r^(p-1) to get the smallest m, when only p is known.
Two cases must be checked:
-> r = 2: if 2^(p-3) + 1 is prime, then m = 2^(p-2) + 1, and
          if 2^(p-3) - 1 is prime, then m = 2^(p-2) - 1.
If there is a solution m in the case r = 2, then a(p) is this smallest solution m (see example a(11)); if there is no solution m with r = 2, then try the 2nd case.
-> r odd prime: if r^(p-1) + 2 is prime, then m = r^(p-1) + 1, and
                if r^(p-1) - 2 is prime, then m = r^(p-1) - 1 (example a(13)).

			tau(2^2 - 1) = 2 = 2*1, so a(1) = 2.
tau(3^2 - 1) = 4 = 2*2, so a(2) = 3.
tau(4^2 - 1) = 4 = 2*2, tau(5^2 - 1) = 8 = 2*4 so a(4) = 5.
For a(11): if r = 2, 2^8 + 1 = 257 is prime, while 2^8 - 1 is not prime, hence a(11) = 2^9 + 1 = 513.
For a(13):
  if r = 2, 2^10 +- 1 are not prime, so not possible;
  if r = 3, 3^12 +- 2 are not prime, so not possible;
  if r = 5, 5^12 +- 2 are not prime, so not possible;
  if r = 7, 7^12 - 2 = 13841287199 is prime, while 7^12 + 2 is not prime, then a(13) = 13841287199+1 = 13841287200.

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, A347191, A347192, A347194.

a(31)-a(35) from Jinyuan Wang, Sep 23 2021

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

Original entry on oeis.org

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 *)

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A347191 Number of divisors of n^2-1.

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A347193 a(n) is the smallest m such that A347191(m) = 2*n, where A347191(m) = tau(m^2 - 1).

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

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