A000197 -id:A000197 - cp's OEIS frontend

A063979 Number of decimal digits in (n!)!; A000197.

1, 1, 1, 3, 24, 199, 1747, 16474, 168187, 1859934, 22228104, 286078171, 3949867548, 58284826485, 915905054360, 15276520209206, 269617872744249, 5021159048900643, 98417586560408168, 2025488254833817394, 43675043585825292775, 984729344827900257489, 23172929656443132617906

Offset: 0

Robert G. Wilson v, Sep 05 2001

Jon E. Schoenfield, Table of n, a(n) for n = 0..448 (first 100 terms from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Factorial

Cf. A000197, A063944, A034886.

// Using about 100 more digits of precision than needed.
nMax:=30; SetDefaultRealField(RealField(Ceiling(Log(10,Factorial(nMax))+100))); a:=[]; for n in [0..nMax] do a[n+1]:=1+Floor(LogGamma(Factorial(n)+1)/Log(10)); end for; a; // Jon E. Schoenfield, Aug 07 2015

seq(length((n)!!), n=0..19); # Zerinvary Lajos, Mar 10 2007

LogBase10Stirling[n_] := Floor[ Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5) + 5246819/(75246796800n^6)]]; (* A001163/A001164; good to at least a(1000) *) LogBase10Stirling[0] = LogBase10Stirling[1] = 0; Table[1 + LogBase10Stirling[n!], {n, 0, 101}] (* Robert G. Wilson v, Aug 05 2015 *)

\\ Using 100 digits of precision.
 a(n)=localprec(100); my(t=n!);return(floor((t*log(t)-t+1/2*log(2*Pi*t)+1/(12*t))/log(10)+1))\\ Robert Gerbicz, Jul 08 2008

More terms from Vladeta Jovovic, Sep 06 2001
A correspondent reported that terms a(17) - a(19) shown here were wrong. That's not true, they are correct. The correspondent was using Python, where the default precision was not large enough to calculate these terms correctly. Thanks to Brendan McKay, Max Alekseyev and Robert Gerbicz for confirming the entries. - N. J. A. Sloane, Jul 08 2008
a(20) from Brendan McKay, Jul 08 2008
a(21)-a(22) from Hugo Pfoertner, Nov 25 2023

A063944 Final nonzero digit of (n!)! (A000197).

Original entry on oeis.org

1, 1, 2, 2, 6, 6, 6, 4, 6, 8, 2, 8, 8, 6, 4, 4, 6, 6, 8, 2, 6, 4, 4, 8, 2, 2, 6, 2, 2, 6, 4, 6, 2, 2, 8, 4, 6, 8, 2, 2, 2, 2, 8, 6, 6, 6, 2, 2, 6, 8, 4, 2, 2, 2, 8, 8, 4, 4, 2, 6, 8, 6, 4, 6, 6, 4, 8, 2, 2, 4, 4, 2, 8, 2, 8, 2, 4, 2, 8, 8, 6, 8, 2, 8, 4, 4, 6, 8, 8, 6, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 2, 6, 4, 2, 8, 2, 4, 2

Offset: 0

Jason Earls, Sep 01 2001

David W. Wilson, Table of n, a(n) for n = 0..10000
Index entries for sequences related to final digits of numbers

Cf. A000197, A008904.

for(n=0,22,m=n!!; while(Mod(m,10) == 0,m=m/10); print(Mod(m,10)))

from functools import reduce
from math import prod, factorial
from sympy.ntheory.factor_ import digits
def A063944(n): return reduce(lambda x,y:x*y%10,((1,1,2,6,4)[a]*((6,2,4,8)[i*a&3] if i*a else 1) for i, a in enumerate(digits(factorial(n),5)[-1:0:-1])))*6%10 if n>1 else 1 # Chai Wah Wu, Dec 07 2023

More terms from David W. Wilson, Sep 05 2001, who remarks that "I'll tell you, computing (107!)! took up some disk space!"

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]

A062274 Number of prime divisors (with repetition) of (n!)!, A000197.

Original entry on oeis.org

0, 0, 1, 7, 45, 291, 2030, 15695, 135045, 1287243, 13495669, 154516663, 1919455487, 25721712601, 369942275033

Offset: 0

Jason Earls, Jul 04 2001

Cf. A000197, A001222, A003604, A022559.

Table[PrimeOmega[(n!)!],{n,0,10}] (* Harvey P. Dale, Apr 29 2015 *)

for(n=0, 10, print1(bigomega(n!!), ", "))

a(n) = { my(res = 0, nf = n!); forprime(p = 2, nf, res+=val(nf, p) ); res }
val(n, p) = my(r=0); while(n, r+=n\=p);r \\ David A. Corneth, Apr 10 2021

from sympy import factorial,factorint
def A062274(n): return sum(sum(factorint(i).values()) for i in range(2,factorial(n)+1)) # Chai Wah Wu, Apr 10 2021

a(n) = A001222(A000197(n)). - Michel Marcus, Oct 20 2019

More terms from David W. Wilson, Jul 06 2001
a(11)-a(13) from Jinyuan Wang, Apr 01 2020
a(14) from David A. Corneth, Apr 10 2021

A036298 (n!)!, n>0 (note that A000197 is the correct version of this sequence, with a(0) = 1, not 0).

Original entry on oeis.org

0, 1, 2, 720, 620448401733239439360000

Offset: 0

N. J. A. Sloane

A006990 Largest prime <= n!.

Original entry on oeis.org

2, 5, 23, 113, 719, 5039, 40289, 362867, 3628789, 39916787, 479001599, 6227020777, 87178291199, 1307674367953, 20922789887947, 355687428095941, 6402373705727959, 121645100408831899, 2432902008176639969, 51090942171709439969, 1124000727777607679927

Offset: 2

N. J. A. Sloane, Robert G. Wilson v

nonn

Conjecture: For n >= 2, n! - a(n) is 1 or a prime, see A033933. - Amarnath Murthy, Mar 19 2002

a(n) is the largest prime divisor of (n!)! of the sequence A000197. - Stanislav Sykora, Jul 14 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 2..400
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Cf. A000142, A007917.

PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; k ]; Table[ PrevPrime[ n! ], {n, 3, 25} ]
Join[{2},NextPrime[Range[3,30]!,-1]] (* Harvey P. Dale, Jan 24 2014 *)

More terms from Jud McCranie; also from Robert G. Wilson v, Jan 03 2001

A058295 Products of distinct factorials.

Original entry on oeis.org

1, 2, 6, 12, 24, 48, 120, 144, 240, 288, 720, 1440, 2880, 4320, 5040, 5760, 8640, 10080, 17280, 30240, 34560, 40320, 60480, 80640, 86400, 103680, 120960, 172800, 207360, 241920, 362880, 483840, 518400, 604800, 725760, 967680, 1036800, 1209600

Offset: 1

Leroy Quet, Dec 07 2000

nonn

(A075082(n)!)^2 is a member for n>0, for example, (6!)^2=6!*5!*3!. Factorials A000142 and superfactorials A000178 (without their first terms), double-superfactorials A098694 and product-of-next-n-factorials A074319 are all subsequences. Products-of-factorials A001013 is a supersequence. - Jonathan Sondow, Dec 18 2004
A000197(n)^2 is a member for n > 2, as ((n!)!)^2 = (n!)!*n!*(n!-1)!. - Jonathan Sondow, Dec 21 2004
Erdős & Graham show that there are exp((1+o(1))n log log n / log n) members of this sequence using no factorials above n.

			288 is included because 288 = 2! * 3! * 4!.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Paul Erdős and Ron L. Graham, On products of factorials, Bull. Inst. Math. Acad. Sinica 4:2 (1976), pp. 337-355.
Index entries for sequences related to factorial numbers

Cf. A075082, A000142, A000178, A098694, A074319, A001013, A000197.

k=10; m=1; With[{p=With[{s=Subsets[Table[n!, {n, 2, k}]]}, Sort[Table[Apply[Times, s[[n]]], {n, Length[s]}]]]}, While[p[[m]]<(k+1)!, m++ ]; Union[Take[p, m-1]]] (* Jonathan Sondow *)

list(lim)=my(v=List([1]),n=1,t=1);while((t=n++!)<=lim,for(i=1,#v,if(v[i]*t<=lim,listput(v,v[i]*t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Mar 26 2012

Corrected by Jonathan Sondow, Dec 18 2004

A003604 Number of primes <= n!.

Original entry on oeis.org

0, 0, 1, 3, 9, 30, 128, 675, 4231, 30969, 258689, 2428956, 25306287, 289620751, 3610490805, 48686912930, 706003798139, 10953617995740, 181035032207760, 3175094503778521, 58893601709552538, 1151825702178908788, 23688535118132456668, 511050155316058710033

Offset: 0

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Number of distinct prime divisors of (n!)!, (A000197). - Jason Earls, Jul 04 2001

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Baugh, Table of n, a(n) for n = 0..25 (a(24)-a(25) found using Kim Walisch's primecount program).
Andrew R. Booker, The Nth Prime Page
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993
R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993
R. G. Wilson v, Letter to N. J. A. Sloane, Jan. 1994
Index entries for sequences related to numbers of primes in various ranges

Cf. A000040, A000197.

Mathematica
```
Table[PrimePi[n!], {n, 0, 16}]
```
PARI
```
for(n=0,10,print(omega(n!!)))
```

a(n)=primepi(n!) \\ Charles R Greathouse IV, Jan 21 2016

[prime_pi(factorial(n)) for n in range(0, 14)] # Zerinvary Lajos, Jun 06 2009

a(15) from Jud McCranie
a(16)-a(17) from Paul Zimmermann
a(18) from Donovan Johnson, Dec 18 2009
a(19) from Donovan Johnson, Feb 18 2010
a(20) from Henri Lifchitz, Nov 11 2012
a(21)-a(23) from Henri Lifchitz, Aug 26 2014

A174228 Divisors of 24!.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95

Offset: 1

Reinhard Zumkeller, Mar 13 2010

24! = 2^22 * 3^10 * 5^4 * 7^3 * 11^2 * 13 * 17 * 19 * 23;
the sequence is finite with A027423(24) = 242880 terms:
a(242880) = A000142(24) = A000197(4) = A010050(12) = 620448401733239439360000 is the last term.
Differs from the infinite sequence A080683 first for 169, 289, 338, 361 etc. - R. J. Mathar, Mar 17 2010

Cf. A018253, A155182.

Select[Divisors[24!],#<=100&] (* Harvey P. Dale, Nov 26 2023 *)

divisors(24!) \\ Charles R Greathouse IV, Sep 02 2015

A152168 Number of binary digits in (n!)!.

Original entry on oeis.org

1, 1, 2, 10, 80, 661, 5802, 54725, 558704, 6178565, 73840164, 950331113, 13121175977, 193618002604, 3042570732326, 50747501675076, 895651186352884, 16679929313440954, 326936145826028780, 6728526339596831313, 145085354333183129464, 3271200076443827203823

Offset: 0

Jon E. Schoenfield, Nov 27 2008

			(3!)! = 6! = 720 has ten binary digits (1011010000), so a(3) = 10.

Jon E. Schoenfield, Table of n, a(n) for n = 0..30
Index entries for sequences related to factorial numbers

Cf. A000197, A072831.

a(n) = length(binary(n!!)) \\ Michel Marcus, Jun 02 2013

cp's OEIS Frontend

A063979 Number of decimal digits in (n!)!; A000197.

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A063944 Final nonzero digit of (n!)! (A000197).

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

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Maple

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A062274 Number of prime divisors (with repetition) of (n!)!, A000197.

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Formula

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A036298 (n!)!, n>0 (note that A000197 is the correct version of this sequence, with a(0) = 1, not 0).

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A006990 Largest prime <= n!.

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Mathematica

Extensions

A058295 Products of distinct factorials.

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Programs

Mathematica

PARI

Extensions

A003604 Number of primes <= n!.

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