A062569 -id:A062569 - cp's OEIS frontend

A027423 Number of divisors of n!.

1, 1, 2, 4, 8, 16, 30, 60, 96, 160, 270, 540, 792, 1584, 2592, 4032, 5376, 10752, 14688, 29376, 41040, 60800, 96000, 192000, 242880, 340032, 532224, 677376, 917280, 1834560, 2332800, 4665600, 5529600, 7864320, 12165120, 16422912

Offset: 0

Glen Burch (gburch(AT)erols.com), Leroy Quet

It appears that a(n+1)=2*a(n) if n is in A068499. - Benoit Cloitre, Sep 07 2002
Because a(0) = 1 and for all n > 0, 2*a(n) >= a(n+1), the sequence is a complete sequence. - Frank M Jackson, Aug 09 2013
Luca and Young prove that a(n) divides n! for n >= 6. - Michel Marcus, Nov 02 2017

			a(4) = 8 because 4!=24 has precisely eight distinct divisors: 1, 2, 3, 4, 6, 8, 12, 24.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Daniel Berend and J. E. Harmse, Gaps between consecutive divisors of factorials, Ann. Inst. Fourier, 43 (3) (1993), 569-583.
Paul Erdős, S. W. Graham, Alexsandr Ivić, and Carl Pomerance, On the number of divisors of n!, Analytic Number Theory, Proceedings of a Conference in Honor of Heini Halberstam, ed. by B. C. Berndt, H. G. Diamond, A. J. Hildebrand, Birkhauser 1996, pp. 337-355.
Florian Luca and Paul Thomas Young, On the number of divisors of n! and of the Fibonacci numbers, Glasnik Matematicki, Vol. 47, No. 2 (2012), 285-293. DOI: 10.3336/gm.47.2.05.
John D. Mahony, The next number in the sequence, Math. Gaz. (2024) Vol. 108, Issue 573, 399-406.
Wikipedia, Complete sequence.
Index entries for sequences related to factorial numbers
Index entries for sequences related to divisors of numbers

Cf. A000005, A000142, A062569, A131688, A161466 (divisors of 10!).

a027423 n = f 1 $ map (\p -> iterate (* p) p) a000040_list where
   f y ((pps@(p:_)):ppss)
     | p <= n = f (y * (sum (map (div n) $ takeWhile (<= n) pps) + 1)) ppss
     | otherwise = y
-- Reinhard Zumkeller, Feb 27 2013
(Python 3.8+)
from math import prod
from collections import Counter
from sympy import factorint
def A027423(n): return prod(e+1 for e in sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values()) # Chai Wah Wu, Jun 25 2022

Maple
```
A027423 := n -> numtheory[tau](n!);
```

Table[ DivisorSigma[0, n! ], {n, 0, 35}]

for(k=0,50,print1(numdiv(k!),", ")) \\ Jaume Oliver Lafont, Mar 09 2009

a(n)=my(s=1,t,tt);forprime(p=2,n,t=tt=n\p; while(tt, t+=tt\=p); s*=t+1); s \\ Charles R Greathouse IV, Feb 08 2013

a(n) <= a(n+1) <= 2*a(n) - Benoit Cloitre, Sep 07 2002
From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 28 2009: (Start)
Assume, p1,p2...pm are the prime numbers less than or equal to n.
Then, a(n) = Product_{i=1..m} (bi+1), where bk = Sum_{i=1..m} floor(n/pk^i).
For example, if n=5, p1=2,p2=3,p3=5;
b1=floor(5/2)+floor(5/2^2)+floor(5/2^3)+...=2+1+0+..=3 similarly, b2=b3=1;
Thus a(5)=(3+1)(1+1)(1+1)=16. (End)
a(n) = A000005(A000142(n)). - Michel Marcus, Sep 13 2014
a(n) ~ exp(c * n/log(n) + O(n/log(n)^2)), where c = A131688 (Erdős et al., 1996). - Amiram Eldar, Nov 07 2020

A079210 Positive divisors of n!, listed in increasing order for each n, a new row for each n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 6, 8, 12, 24, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

This sequence is the same as A070861 for the first 38 terms, but differs thereafter.

			First few rows are:
1;
1;
1,2;
1,2,3,6;
1,2,3,4,6,8,12,24;
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120;
...

Andrew Howroyd, Table of n, a(n) for n = 0..1979 (rows 0..12)

Cf. A027423 (row lengths), A062569 (row sums), A070861.

[Divisors(Factorial(n)): n in [0..10]]; // Vincenzo Librandi, Jun 19 2015

Flatten[Table[Divisors[n!],{n,6}]]  (* Harvey P. Dale, Mar 13 2011 *)

tabf(nn) = for (n=0, nn, print(divisors(n!))); \\ Michel Marcus, Jun 19 2015

a(0)=1 prepended by Andrew Howroyd, Jan 26 2022

A068499 Numbers m such that m! reduced modulo (m+1) is not zero.

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 52, 58, 60, 66, 70, 72, 78, 82, 88, 96, 100, 102, 106, 108, 112, 126, 130, 136, 138, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232, 238, 240, 250, 256, 262, 268, 270

Offset: 1

Benoit Cloitre, Mar 11 2002

Also n such that tau((n+1)!) = 2* tau(n!)
For n > 2, a(n) is the smallest number such that a(n) !== -1 (mod a(k)+1) for any 1 < k < n. [Franklin T. Adams-Watters, Aug 07 2009]
Also n such that sigma((n+1)!) = (n+2)* sigma(n!), which is the same as A062569(n+1) = (n+2)*A062569(n). - Zak Seidov, Aug 22 2012
This sequence is obtained by the following sieve: keep 1 in the sequence and then, at the k-th step, keep the smallest number, x say, that has not been crossed off before and cross off all the numbers of the form k*(x + 1) - 1 with k > 1. The numbers that are left form the sequence. - Jean-Christophe Hervé, Dec 12 2015
a(n) = A039915(n-1) for 3 < n <= 1000. - Georg Fischer, Oct 19 2018

			Illustration of the sieve: keep 1 = a(1) and then
1st step: take 2 = a(2) and cross off 5, 8, 11, 14, 17, 20, 23, 26, etc.
2nd step: take 3 = a(3) and cross off 7, 11, 15, 19, 23, 27, etc.
3rd step: take 4 = a(4) and cross off 9, 14, 19, 24, etc.
4th step: take 6 = a(5) and cross off 13, 19, 25 etc.
10 is obtained at next step and the smallest crossed off numbers are then 21 and 28. This gives the beginning of the sequence up to 22 = a(10): 1, 2, 3, 4, 6, 10, 12, 16, 18, 22. - _Jean-Christophe Hervé_, Dec 12 2015

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000
David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Remark 3.3, p. 6.
Karlee J. Westrem, Schaper numbers, palindrome partitions, and symmetric functions, with applications to characters of the symmetric group, Ph. D. Dissertation, Michigan Tech. Univ. (2025). See p. 49.
Index entries for sequences generated by sieves

Cf. A000040, A039915, A062569, A166460 (almost complement).

Select[Range[300],Mod[#!,#+1]!=0&] (* Harvey P. Dale, Apr 11 2012 *)

{plnt=1 ; nfa=1; mxind=60 ;  for(k=1, 10^7, nfa=nfa*k;
if(nfa % (k+1) != 0 , print1(k, ", "); plnt++ ;
if(mxind <  plnt, break() )))} \\ Douglas Latimer, Apr 25 2012

a(n)=if(n<5,n,prime(n-1)-1) \\ Charles R Greathouse IV, Apr 25 2012

from sympy import prime
def A068499(n): return prime(n-1)-1 if n>3 else n # Chai Wah Wu, Aug 27 2024

For n >= 4, a(n) = prime(n-1) - 1 = A006093(n-1).
For n <> 3, all terms are one less prime. - Zak Seidov, Aug 22 2012
a(n) = Integer part of A078456(n+1)/A078456(n). - Eric Desbiaux, May 07 2013

A078156 A078153(n!).

Original entry on oeis.org

0, 0, 0, 0, 46, 702, 7479, 97902, 1231886, 15977798, 208298944, 3085485116, 45879947392, 749485746579, 12963973882204, 236404256556347, 4415737043058504, 88721524940832020, 1830113429944169943, 40228564066847381090, 921832573196324390682

Offset: 1

Labos Elemer, Nov 27 2002

nonn

Cf. A078152, A078153, A078154, A078155, A078156, A078157, A078158.

Table[Length[Union[Table[Floor[w!/j], {j, 1, w!}]]] -DivisorSigma[1, w! ], {w, 1, 9}]

a(n) = A078153(n!) = A078162(n) - A062569(n)

Terms a(10) onward from Max Alekseyev, Feb 12 2012

A167367 a(n) = sigma(n!!) where n!! is A006882(n).

Original entry on oeis.org

1, 1, 3, 4, 15, 24, 124, 192, 1020, 1920, 12264, 23040, 159666, 322560, 2555280, 5041344, 40893840, 90744192, 761260368, 1814883840, 15732804296, 38900010240, 377587663200, 933600245760, 9087075973248, 23520702965760, 254438142416640

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 01 2009

nonn

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

Cf. A062569.

Join[{1},Array[DivisorSigma[1,#!! ]&,50,1]]

a(n)=sigma(prod(i=0, (n-1)\2, n - 2*i )) \\ Charles R Greathouse IV, May 01 2016

a(n) = A000203(A006882(n)). - R. J. Mathar, Feb 07 2011

A366757 a(n) is the sum of the divisors of n!-1.

Original entry on oeis.org

1, 6, 24, 144, 720, 5040, 42096, 399000, 3753960, 47500992, 479001600, 6230615736, 87178291200, 1457696910960, 20929670124480, 379536693283440, 6510917252872320, 121831439598033840, 2432921507427445440, 53921727651043042560, 1134312679767378217920

Offset: 2

Sean A. Irvine, Oct 20 2023

nonn

			a(5) = 144 because the divisors of 5!-1 are {1, 7, 17, 119}.

Cf. A033312, A000203, A064145, A062569, A366758.

a:=n->numtheory[sigma](n!-1):
seq(a(n), n=2..30);

DivisorSigma[1,Range[2,25]!-1] (* Paolo Xausa, Oct 21 2023 *)

from math import factorial
from sympy import divisor_sigma
def A366757(n): return divisor_sigma(factorial(n)-1) # Chai Wah Wu, Oct 20 2023

a(n) = sigma(n!-1) = A000203(A033312(n)).

A061556 a(n) is the least k > 0 such that sigma(k!) >= n*k!.

Original entry on oeis.org

1, 1, 3, 5, 9, 14, 23, 43, 79, 149, 263, 461, 823, 1451, 2549, 4483, 7879, 13859, 24247, 42683, 75037, 131707, 230773, 405401, 710569, 1246379, 2185021, 3831913, 6720059, 11781551, 20657677

Offset: 0

Labos Elemer, May 17 2001

It seems that, for n > 1, a(n+1) < 2*a(n). Does lim_{n -> infinity} a(n+1)/a(n) = 2? - Benoit Cloitre, Aug 18 2002
Smallest number m such that the abundancy-index of m! is at least n.
Floor(sigma(m!)/m!) = n; note that abundancy-index [= sigma(u)/u] here is not necessarily an integer.
It appears that a(n) = A091440(n) for n >= 13. - Daniel Suteu, Sep 03 2019

			floor(sigma(842!)/842!) = 11 while floor(sigma(843!)/843!) = 12.

Achim Flammenkamp, The Multiply Perfect Numbers Page.
Fred Helenius, Multiperfect numbers.

Cf. A000142, A000203, A023199, A062569.

PARI
```
a(n)=if(n<0,0,s=1; while(sigma(s!)
				
```

a(n) = Min{w | floor(sigma(w!)/w!) = n}.

More terms from David Wasserman, Jun 18 2002

a(1) inserted and a(21)-a(30) added by Daniel Suteu, Sep 03 2019

A064028 Sum of the unitary divisors of n!.

Original entry on oeis.org

1, 3, 12, 36, 216, 1020, 8160, 61920, 507744, 4383392, 52600704, 624249600, 8739494400, 109190390400, 1583122968000, 25318378008000, 455730804144000, 8193040840252800, 163860816805056000, 3256371347261760000, 67204676251838361600, 1366492477414792734720

Offset: 1

Labos Elemer, Sep 11 2001

nonn

			n=6, 6! = 720, sum of the 8 unitary ones of its 30 divisors is 1020, a(6) = 720+1+16+45+9+80+5+144 = 1020.

Amiram Eldar, Table of n, a(n) for n = 1..450
Charles R. Wall, Problem H-374, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 22, No. 3 (1984), p. 280; Bounds of Joy, Solution to Problem H-374 by the proposer, ibid., Vol. 24, No. 2 (1986), p. 188.

Cf. A034448, A046656, A056657, A056171, A056172, A000203, A000142, A062569.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma/@ (Range[17]!) (* Amiram Eldar, Jun 23 2019 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1); forprime(p=2,n, s*=p^valp(n,p)+1); s \\ Charles R Greathouse IV, Jan 26 2023

a(n) = usigma(n!) = A034448(A000142(n)).

a(n)/n! <= 2 (while usigma(n)/n and sigma(n!)/n! are unbounded; Wall, 1984). - Amiram Eldar, Feb 08 2022

A104353 Sum of divisors of A104350(n).

Original entry on oeis.org

1, 3, 12, 28, 168, 546, 4368, 9360, 28800, 148800, 1785600, 5401440, 75620160, 538793640, 2711348640, 5603453856, 100862169408, 303420079872, 6068401597440, 30380907997440, 213199354368000, 2362959510912000, 56711028261888000, 170288884313856000

Offset: 1

Reinhard Zumkeller, Mar 06 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..640
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000203, A104350, A104352, A062569.

DivisorSigma[1, FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 30]]] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); p[#p];}
a(n) = sigma(prod(k = 2, n, gpf(k))); \\ Amiram Eldar, Apr 08 2024

a(n) = A000203(A104350(n)).

a(14), a(21) corrected by Georg Fischer, Feb 28 2023

A153824 Sum of proper divisors of n!: a(n) = sigma(n!) - n!.

Original entry on oeis.org

0, 0, 1, 6, 36, 240, 1698, 14304, 118800, 1118160, 11705288, 144092256, 1738439808, 24817158912, 355309325280, 5378578601760, 86081749397280, 1570394279039040, 28281459220193088, 572031558109589760, 11458497230555094720

Offset: 0

Omar E. Pol, Jan 02 2009

nonn

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

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

Cf. A000142, A000203, A001065, A062569.

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

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

Table[DivisorSigma[1, n!] - n!, {n, 0, 50}] (* G. C. Greubel, Aug 30 2016 *)

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

a(n) = A000203(n!) - n! = A062569(n) - A000142(n) = A001065(A000142(n)).

Extended by Emeric Deutsch, Jan 07 2009

cp's OEIS Frontend

A027423 Number of divisors of n!.

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A079210 Positive divisors of n!, listed in increasing order for each n, a new row for each n.

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A068499 Numbers m such that m! reduced modulo (m+1) is not zero.

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A078156 A078153(n!).

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A167367 a(n) = sigma(n!!) where n!! is A006882(n).

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A366757 a(n) is the sum of the divisors of n!-1.

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A061556 a(n) is the least k > 0 such that sigma(k!) >= n*k!.

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