A064144 -id:A064144 - cp's OEIS frontend

A064145 a(n) = tau(n!-1) or number of divisors of n!-1.

1, 2, 2, 4, 2, 2, 4, 6, 4, 16, 2, 4, 2, 24, 4, 8, 8, 8, 4, 16, 8, 4, 4, 8, 4, 4, 16, 32, 2, 8, 2, 2, 4, 8, 4, 32, 2, 16, 4, 16, 16, 128, 16, 32, 32, 4, 16, 8, 4, 32, 32, 16, 64, 64, 32, 64, 32, 4, 8, 16, 16, 32, 16, 64, 16, 128, 4, 64, 32, 32, 8, 16, 32, 128, 8

Offset: 2

Vladeta Jovovic, Sep 11 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 2..135
Hisanori Mishima, Factorizations of many number sequences

Cf. A000005, A002582, A027423, A002981, A002982, A064144.

Do[ Print[ DivisorSigma[0, n! - 1]], {n, 2, 40} ]
DivisorSigma[0,Range[2,80]!-1] (* Harvey P. Dale, Aug 17 2024 *)

{ f=1; for (n=2, 100, f*=n; if (n>1, a=numdiv(f - 1), a=0); write("b064145.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 09 2009

More terms from Robert G. Wilson v, Oct 04 2001
a(51)-a(76) from Harry J. Smith, Sep 09 2009
Ambiguous term a(1) removed by Max Alekseyev, May 06 2022

A104369 Number of divisors of A104365(n) = A104350(n) + 1.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 4, 4, 16, 16, 4, 2, 16, 4, 4, 8, 4, 2, 16, 8, 8, 2, 8, 16, 32, 16, 32, 16, 8, 8, 16, 128, 4, 32, 16, 16, 4, 16, 8, 16, 4, 8, 128, 16, 4, 16, 4, 8, 4, 16, 8, 16, 32, 8, 16, 16, 4, 2, 4, 16, 64, 16, 8, 8, 16, 16, 64, 64, 64, 32, 8, 64, 16, 32, 8, 8, 32, 8, 16, 8, 16, 128, 64, 16, 8, 16, 4, 8

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Max Alekseyev, Table of n, a(n) for n = 1..158 (terms 142..151 from Tyler Busby)
Reinhard Zumkeller, Products of largest prime factors of numbers <= n

Cf. A000005, A064144, A104350, A104361, A104365, A104366, A104367, A104368, A104370, A104371, A104372.

a[n_] := DivisorSigma[0, 1 + Product[FactorInteger[k][[-1, 1]], {k, 1, n}]]; Array[a, 76] (* Amiram Eldar, Feb 12 2020 *)

a(n) = A000005(A104365(n)).

a(51)-a(76) from Amiram Eldar, Feb 12 2020

Terms a(77) onward from Max Alekseyev, Oct 03 2022

A344859 a(n) is the number of divisors of n^n + 1.

Original entry on oeis.org

2, 2, 2, 6, 2, 8, 8, 16, 8, 16, 8, 96, 16, 32, 48, 160, 4, 12, 288, 48, 8, 64, 16, 512, 64, 128, 32, 3072, 64, 128, 1024, 384, 16, 2048, 64, 18432, 32, 128, 192, 512, 768, 64, 1024, 384, 256, 16384, 256, 2560, 64, 192, 1024, 3072, 32, 512, 16384, 4096, 128, 8192, 8192, 768, 4096, 256, 128, 1376256, 16

Offset: 0

Seiichi Manyama, May 31 2021

nonn

Tyler Busby, Table of n, a(n) for n = 0..148 (terms 0..123 from Max Alekseyev)
Eric Weisstein's World of Mathematics, Sierpinski Number of the First Kind

Cf. A000005, A014566, A062319, A064144, A121270, A334167.

Cf. A007571 (Lpf), A055385 (lpf).

a[0] = 2; a[n_] := DivisorSigma[0, n^n + 1]; Array[a, 45, 0] (* Amiram Eldar, May 31 2021 *)

PARI
```
a(n) = numdiv(n^n+1);
```

a(n) = A000005(A014566(n)).

A366758 a(n) is the sum of the divisors of n!+1.

Original entry on oeis.org

3, 3, 4, 8, 31, 133, 832, 5113, 41044, 388800, 3958704, 39916802, 518682390, 6302045232, 90968651712, 1332614649600, 22844265373440, 356226551466344, 7504470340300800, 123358411682195904, 2432902126073962432, 52279222588118377280, 1175121515279802150144

Offset: 0

Sean A. Irvine, Oct 20 2023

nonn

			a(5) = 133 because the divisors of 5!+1 are {1, 11, 121}.

Cf. A038507, A000203, A064144, A062569, A366757.

a:=n->numtheory[sigma](n!+1):
seq(a(n), n=0..30);

DivisorSigma[1,Range[0,25]!+1] (* Paolo Xausa, Oct 21 2023 *)

from math import factorial
from sympy import divisor_sigma
def A366758(n): return divisor_sigma(factorial(n)+1) # Chai Wah Wu, Oct 20 2023

a(n) = sigma(n!+1) = A000203(A038507(n)).

cp's OEIS Frontend

A064145 a(n) = tau(n!-1) or number of divisors of n!-1.

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A104369 Number of divisors of A104365(n) = A104350(n) + 1.

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A344859 a(n) is the number of divisors of n^n + 1.

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A366758 a(n) is the sum of the divisors of n!+1.

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