A347193 -id:A347193 - 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

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 *)

A347194 Numbers such that the two adjacent integers are a prime and the square of another prime.

Original entry on oeis.org

8, 10, 24, 48, 168, 360, 840, 1368, 1848, 2208, 3720, 5040, 7920, 10608, 11448, 16128, 17160, 19320, 29928, 36480, 44520, 49728, 54288, 57120, 66048, 85848, 97968, 113568, 128880, 177240, 196248, 201600, 218088, 241080, 273528, 292680, 323760, 344568, 368448, 426408, 458328, 516960, 528528, 537288, 552048, 564000, 573048, 579120

Offset: 1

Bernard Schott, Sep 23 2021

nonn

-> Equivalently, numbers k such that tau(k^2-1) = A347191(k) = 6 (see example; used for Maple code).
Proof: tau(k^2-1) = 6 <==> k^2-1 = p^5 or k^2-1 = p*q^2 with p <> q primes; but k^2-p^5 = 1 is impossible, as a consequence of the Catalan-Mihăilescu theorem; now, (k-1)*(k+1) = p*q^2 ==> (k-1 = p and k+1 = q^2) or (k-1 = q^2 and k+1 = p), because k-1 = q and k+1 = p*q is not possible, otherwise 2 = q*(p-1), which would contradict p <> q.
-> There are two possible configurations with p, q primes: (q^2 < a(n) < p) or (p < a(n) < q^2).
The unique configuration q^2 < a(n) < p is for q = 3, a(2) = 10 and  p = 11.
All the other configurations, for n = 1 or n >= 3, are of the form p < a(n) < q^2 with p = A049002(n) and q = A062326(n).
-> Note that there is only one integer such that the two adjacent integers are a prime and the square of that prime: it is 3, which lies between 2 and 2^2; in this case, tau(3^2-1) = 4.

			8 is a term since 8 lies between 7 (prime) and 9 = 3^2 (square of prime); also tau(8^2-1) = tau(63) = 6.
10 is a term since 10 lies between 9 = 3^2 (square of prime) and 11 (prime); also tau(10^2-1) = tau(99) = 6.
24 is a term since 24 lies between 23 (prime) and 25 = 5^2 (square of prime); also tau(24^2-1) = tau(575) = 6.

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.
Wikipedia, Catalan's conjecture.

Cf. A049002, A062326, A146981.
Cf. A347191, A347192, A347193.
Subsequence of A163492 (between prime and a perfect square).

with(numtheory):
filter := q-> tau(q^2-1) = 6 : select(filter, [$2..580000]);

q[n_] := Module[{e1 = FactorInteger[n - 1][[;; , 2]], e2 = FactorInteger[n + 1][[;; , 2]]}, (e1 == {1} && e2 == {2}) || (e1 == {2} && e2 == {1})]; Select[Range[4, 600000], q] (* Amiram Eldar, Sep 23 2021 *)

isok(m) = my(pa, pb); (isprimepower(m-1, &pa)*isprimepower(m+1, &pb) == 2) && (pa != pb); \\ Michel Marcus, Sep 23 2021

upto(n) = { my(res = List()); forprime(i = 3, sqrtint(n-1), if(isprime(i^2 - 2), listput(res, i^2-1); ); if(isprime(i^2 + 2), listput(res, i^2 + 1); ) ); res } \\ David A. Corneth, Sep 23 2021

For n >= 3: a(n) = A049002(n) + 1 = a(n) = A146981(n) - 1 = (A049002(n) + A146981(n))/2 = A062326(n)^2 - 1.

A353008 a(n) is the smallest positive k such that k^2 + 1 has 2*n divisors, or -1 if no such k exists.

Original entry on oeis.org

1, 3, 7, 13, 182, 43, 1068, 47, 268, 443, 15905182, 157, 1832311432, 14557, 16432, 307, 255250280182, 1407, 355101282318, 3307, 92682, 3626068, 21346690797155182, 993, 313932, 120813568, 51982, 16693, 982692130687379186432, 2943, 2444574943897581751068, 2163

Offset: 1

Jon E. Schoenfield, May 15 2022

nonn

From Jon E. Schoenfield, Jun 14 2024: (Start)
For integers k, neither 3 nor 4 ever divides k^2 + 1, so there exists no prime p < 5 such that p^2 divides k^2 + 1.
For n <= 32, the only n for which the 5-adic valuation of a(n)^2 + 1 is not gpf(n) - 1 is n = 16 (see Examples).
Conjecture: a(n) is never -1. (End)

			From _Jon E. Schoenfield_, Jun 14 2024: (Start)
From a(5) = 182 because 182 is the smallest positive integer k such that k^2 + 1 has 2*5 divisors: 182^2 + 1 = 33125 = 5^4 * 53.
a(16) = 307 because 307 is the smallest positive integer k such that k^2 + 1 has 2*16 divisors: 307^2 + 1 = 94250 = 2 * 5^3 * 377.
a(31) = 2444574943897581751068: 2444574943897581751068^2 + 1 = 5975946656331864965715445578098297119140625 = 5^30 * 6416623862896477837609. (End)

Cf. A006530, A112765, A193432, A347193.

a(26), a(29), and a(31) corrected by Jon E. Schoenfield, Jun 14 2024

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A347191 Number of divisors of n^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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Mathematica

A347194 Numbers such that the two adjacent integers are a prime and the square of another prime.

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Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A353008 a(n) is the smallest positive k such that k^2 + 1 has 2*n divisors, or -1 if no such k exists.

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