A062569 -id:A062569 - cp's OEIS frontend

A153825 Sum of proper divisors minus the number of proper divisors of n!.

0, 0, 0, 3, 29, 225, 1669, 14245, 118705, 1118001, 11705019, 144091717, 1738439017, 24817157329, 355309322689, 5378578597729, 86081749391905, 1570394279028289, 28281459220178401, 572031558109560385, 11458497230555053681

Offset: 0

Omar E. Pol, Jan 02 2009

nonn

a(n) is the sum of proper divisors minus the number of proper divisors of factorial number A000142(n).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000005, A000142, A000203, A001065, A032741, A062569, A152770, A153823, A153824.

[DivisorSigma(1,Factorial(n)) - Factorial(n) - (DivisorSigma(0,Factorial(n))-1): n in [0..20]]; // Vincenzo Librandi, Aug 31 2016

with(numtheory): seq(sigma(factorial(n))-factorial(n)-tau(factorial(n))+1, n = 0 .. 22); # Emeric Deutsch, Jan 07 2009

Table[DivisorSigma[1,n!]-n!-(DivisorSigma[0,n!]-1),{n,0,20}] (* Harvey P. Dale, Jan 14 2012 *)

a(n) = (sigma(n!) - n!) - (numdiv(n!) - 1); \\ Michel Marcus, Aug 31 2016

a(n) = A153824(n) - A153823(n) = A152770(A000142(n)).

Extended by Emeric Deutsch, Jan 07 2009

A167368 a(n) = sigma((n!!)!).

Original entry on oeis.org

1, 1, 3, 12, 159120, 6686252969760, 89050715142739003008099466232718435351438398888454549774336000

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 01 2009

Cf. A062569, A167367.

Mathematica
```
Array[DivisorSigma[1,#!!! ]&,7,1]
```

A167368(n)=sigma( prod(k=0,n\2-1,n-2*k) ! )  \\ M. F. Hasler, Feb 07 2011

a(n) = A000203(A161563(n)).

a(0)=1 prepended by Alois P. Heinz, Jan 12 2022

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)).

A061555 Integer part of sigma(n!)/n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Labos Elemer, May 17 2001

nonn

With increasing n, a(n) goes to infinity (proof in Sierpiński).
From Bernard Schott, Oct 03 2022: (Start)
It seems that sigma(n!)/n! is an integer only for n = 0, 1, 3, 5 and corresponding values are 1, 1, 2, 3.
For m >= 2, the smallest integer n such that a(n) = m is A061556(m). (End)

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 6, p. 169, Warsaw, 1964.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A000142, A000203, A061556, A062569.

Table[Floor[DivisorSigma[1, n!]/n!], {n, 0, 100}] (* Wesley Ivan Hurt, Apr 16 2024 *)

{ for (n=0, 1000, write("b061555.txt", n, " ", sigma(n!)\n!) ) } \\ Harry J. Smith, Jul 24 2009

a(n) = floor(sigma(n!)/n!) = floor(A062569(n)/A000142(n)).

Terms corrected for an offset of 0 by Harry J. Smith, Jul 24 2009

A062008 Number of divisors of (n!)!, or A000197.

Original entry on oeis.org

1, 1, 2, 30, 242880, 23565900177211392000, 2773739201349556936377871973938118055565107020522751759201737480601600000000000000

Offset: 0

Jason Earls, Jul 04 2001

Harry J. Smith, Table of n, a(n) for n=0,...,7

Cf. A000197, A000203, A062569

[NumberOfDivisors(Factorial(Factorial(n))): n in [0..7]]; // Vincenzo Librandi, Nov 09 2014

with(numtheory): A062008:=n->tau((n!)!): seq(A062008(n), n=0..6); # Wesley Ivan Hurt, Nov 08 2014

Table[DivisorSigma[0, (n!)!], {n, 0, 6}] (* Wesley Ivan Hurt, Nov 08 2014 *)

PARI
```
for(n=0,6,print(numdiv(n!!)))
```

{ for (n=0, 7, write("b062008.txt", n, " ", numdiv(n!!)) ) } \\ [Harry J. Smith, Jul 29 2009]

A064138 Sum of non-unitary divisors of n!.

Original entry on oeis.org

0, 0, 0, 24, 144, 1398, 11184, 97200, 973296, 10950696, 131408352, 1593191808, 22304685312, 333297226080, 5103130001760, 81686161277280, 1470350902991040, 26490792085668288, 529815841713365760, 10635027891469974720

Offset: 1

Labos Elemer, Sep 11 2001

nonn

			For n = 6, 6! = 720, the sum of its 30 divisors is 2418, the sum of the 8 unitary divisors is 1020, so the remaining 22 divisors give a(6) = 1398.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A034448, A048105, A046656, A056657, A056171, A056172, A000203, A000142, A062569, A063955, A063960.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - Times @@ f2 @@@ fct; a[1] = 0; Array[a, 20] (* Amiram Eldar, Apr 01 2024 *)

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; f=1; for (n=1, 100, f*=n; write("b064138.txt", n, " ", sigma(f) - usigma(f)); ) } \\ Harry J. Smith, Sep 08 2009

a(n) = sigma(n!) - usigma(n!) = A000203(n!) - A034448(A000142(n)) = A062569(n) - A034448(n!) = A048105(n!).

Term corrected and more terms added by Harry J. Smith, Sep 08 2009

A073081 Greatest k such that k! divides sigma(n!).

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 4, 6, 6, 2, 4, 4, 4, 7, 7, 7, 7, 4, 9, 7, 8, 8, 11, 8, 8, 10, 10, 9, 10, 13, 14, 13, 6, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 8, 11, 11, 12, 19, 21, 16, 16, 16, 14, 14, 12, 16, 16, 16, 16, 16, 16, 22, 21, 21, 21, 19, 19, 20, 20, 20, 20, 22, 21, 21, 19, 14

Offset: 1

Benoit Cloitre, Aug 17 2002

sum(k=1,n,a(k)) seems to be asymptotic to C*n^2 with 1/20 < C < 1/10.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A062569 (sigma(n!)).

a(n) = my(m=1, s=sigma(n!)); for (k=1, n, if ((s % k!) == 0, m = max(m, k))); m; \\ Michel Marcus, Dec 05 2019

A078118 n+1 does not divide sigma(n!).

Original entry on oeis.org

1, 6, 22, 25, 28, 30, 40, 42, 46, 52, 58, 60, 68, 70, 72, 78, 87, 88, 96, 98, 102, 105, 106, 112, 114, 122, 126, 128, 130, 133, 138, 145, 148, 150, 156, 162, 166, 172, 178, 182, 190, 192, 196, 198, 212, 213, 217, 218, 222, 225, 226, 228, 232, 238, 240, 250, 262

Offset: 1

Benoit Cloitre, Dec 05 2002

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..8822
Vaclav Kotesovec, Plot of a(n)/n for n = 1..8000

Cf. A062569.

Select[Range[300], Not[Divisible[DivisorSigma[1, #!], # + 1]] &] (* Vaclav Kotesovec, Feb 16 2019 *)

a(n) seems to be asymptotic to c*n with c=4.8....

A116906 Sum of squares of divisors of n!.

Original entry on oeis.org

1, 1, 5, 50, 850, 22100, 806806, 40340300, 2584263500, 209609328500, 20993420690550, 2561197324247100, 368819285671473000, 62699278564150410000, 12294076739210974071000, 2766341857504878226501200

Offset: 0

Jonathan Vos Post, Mar 15 2006

A262812 Values of n such that sigma(n!) mod sigma(n) is not 0.

Original entry on oeis.org

4, 6, 9, 10, 16, 25, 45, 48, 50, 64, 86, 122, 192, 256, 289, 314, 326, 448, 562, 578, 706, 722, 729, 794, 842, 1226, 1346, 1458, 1514, 1681, 1754, 1922, 2186, 2401, 2566, 2601, 2916, 3362, 3481, 3866, 3986, 4046, 4096, 4274, 4852, 5043, 5186, 5414

Offset: 1

Altug Alkan, Oct 03 2015

nonn

Subsequence of A002808.
Motivation was the investigation of sum of divisors of n! in terms of sum of divisors of n.
Obviously, a(n) cannot be a prime number, although it can be a semiprime number.
Is this sequence infinite?

			a(1) = 4 because sigma(4!) mod sigma(4) = 60 mod 7 = 4.
a(2) = 6 because sigma(6!) mod sigma(6) = 2418 mod 12 = 6.
a(3) = 9 because sigma(9!) mod sigma(9) = 1481040 mod 13 = 2.

Cf. A062569, A000203.

for(n=1, 1e30, if( sigma(n!) % sigma(n) != 0 , print1(n", ")))

A066247(a(n)) = 1.
A000005(a(n)) > 2.
A001222(a(n)) > 1.

cp's OEIS Frontend

A153825 Sum of proper divisors minus the number of proper divisors of n!.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A167368 a(n) = sigma((n!!)!).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A061555 Integer part of sigma(n!)/n!.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A062008 Number of divisors of (n!)!, or A000197.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

A064138 Sum of non-unitary divisors of n!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A073081 Greatest k such that k! divides sigma(n!).

Views

Author

Keywords

Comments

Links

Crossrefs