A000197 -id:A000197 - cp's OEIS frontend

A336686 Decimal expansion of Sum_{k>=0} 1/(k!)!.

2, 5, 0, 1, 3, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 0, 5, 0, 0, 6, 2, 6, 4, 5, 9, 9, 8, 5, 0, 0, 7, 2, 3, 7, 9, 3, 7, 6, 0, 2, 1, 9, 3, 6, 9, 0, 0, 6, 0, 6, 9, 0, 2, 1, 3, 6, 1, 4, 9, 9, 1, 4, 9, 6, 1, 1, 6, 8, 6, 2, 4, 1, 8, 1, 7, 8, 9, 1, 9, 8, 9, 9, 3, 3, 9, 4, 3, 7, 4, 1, 5, 7, 4, 4, 1

Offset: 1

Daniel Hoyt, Nov 20 2020

The sum has 18 8's in a row within the first 23 decimal places.

1801/720 approximates this constant to 23 significant digits.

			2.5013888888888888888888905006...

Cf. A000197, A336810.

evalf(sum(1/(k!)!, k=0..infinity), 112);  # Alois P. Heinz, Nov 20 2020

RealDigits[Sum[1/(k!)!, {k, 0, 4}], 10, 100][[1]] (* Amiram Eldar, Nov 21 2020 *)

PARI
```
suminf(k=0, 1/(k!)!)
```

A245087 Largest number such that 2^a(n) is a divisor of (n!)!.

Original entry on oeis.org

0, 0, 1, 4, 22, 116, 716, 5034, 40314, 362874, 3628789, 39916793, 479001588, 6227020788, 87178291188, 1307674367982, 20922789887982, 355687428095978, 6402373705727977, 121645100408831983, 2432902008176639978, 51090942171709439975, 1124000727777607679972

Offset: 0

Stanislav Sykora, Jul 15 2014

nonn

Also the number of trailing zeros in the binary expansion of (n!)!.

			a(4)=22 because (4!)!=620448401733239439360000 is divisible by 2^22 but not by 2^23.

Stanislav Sykora, Table of n, a(n) for n = 0..399

Cf. A000120 (Hamming weights), A000142, A000197, A011371.

PARI
```
a(n) = n!-hammingweight(n!)
```

a(n) = n! - Hw(n!), Hw being the Hamming weight function.

a(n) = A011371(A000142(n)).

A301861 a(n) is the sum of the decimal digits of (n!)!.

Original entry on oeis.org

1, 1, 2, 9, 81, 783, 7164, 69048, 711009, 7961040, 95935761, 1242436185, 17235507996

Offset: 0

Jon E. Schoenfield, Mar 28 2018

Presumably, lim_{n->oo} a(n)/A008906(n!) = 9/2.

			a(0) = digitsum((0!)!) = digitsum(1!) = digitsum(1) = 1.
a(1) = digitsum((1!)!) = digitsum(1!) = digitsum(1) = 1.
a(2) = digitsum((2!)!) = digitsum(2!) = digitsum(2) = 2.
a(3) = digitsum((3!)!) = digitsum(6!) = digitsum(720) = 7+2 = 9.
a(4) = digitsum((4!)!) = digitsum(24!) = digitsum(620448401733239439360000) = 6+2+0+4+4+8+4+0+1+7+3+3+2+3+9+4+3+9+3+6+0+0+0+0 = 81.

Index entries for sequences related to factorial numbers

Cf. A000142 (factorial numbers), A000197 ((n!)!), A004152 (sum of digits of n!), A007953 (sum of digits of n), A008906 (number of digits in n! excluding trailing zeros), A027868 (number of trailing zeros in n!), A034886 (number of digits in n!), A063979 (number of digits in (n!)!).

[&+Intseq(Factorial(Factorial(n))): n in [0..10]]; // Vincenzo Librandi, Mar 29 2018

a:= n-> add(i, i=convert(n!!, base, 10)):
seq(a(n), n=0..8);  # Alois P. Heinz, Oct 27 2021

Table[Plus@@IntegerDigits[(n!)!], {n, 0, 10}] (* Vincenzo Librandi, Mar 29 2018 *)

a(n) = sumdigits(n!!); \\ Michel Marcus, Mar 28 2018

from math import factorial
def A301861(n):
    return sum(int(d) for d in str(factorial(factorial(n)))) # Chai Wah Wu, Mar 31 2018
# faster program for larger values of n
from gmpy2 import mpz, digits, fac
def A301861(n): return int(sum(mpz(d) for d in digits(fac(fac(n))))) # Chai Wah Wu, Oct 24 2021

a(n) = A007953(A000197(n)). - Michel Marcus, Mar 28 2018

a(n) = A004152(A000142(n)). - Altug Alkan, Mar 28 2018

a(11) from Chai Wah Wu, Mar 31 2018

a(12) from Chai Wah Wu, Apr 01 2018

A348651 Number of ones in the binary expansion of (n!)!.

Original entry on oeis.org

1, 1, 1, 4, 29, 293, 2566, 24844, 259437, 2908263, 35102629, 455204360, 6321171774

Offset: 0

Alois P. Heinz, Oct 27 2021

			a(3) = 4 because (3!)! = 6! = 720 = 1011010000_2 which has 4 ones.

Index entries for sequences related to factorial numbers

Cf. A000120, A000142, A000197, A079584, A152168, A301861.

a:= n-> add(i, i=Bits[Split](n!!)):
seq(a(n), n=0..10);

a[n_] := DigitCount[(n!)!, 2, 1]; Array[a, 10, 0] (* Amiram Eldar, Oct 29 2021 *)

a(n) = hammingweight((n!)!); \\ Michel Marcus, Oct 29 2021

from gmpy2 import fac, popcount
def A348651(n): return popcount(fac(fac(n))) # Chai Wah Wu, Oct 28 2021

a(n) = A000120(A000197(n)).

a(11)-a(12) from Chai Wah Wu, Oct 28 2021

A033187 a(n) = (n!)!/n!.

Original entry on oeis.org

1, 1, 1, 120, 25852016738884976640000

Offset: 0

Ken Alverson (KenA(AT)tso.cin.ix.ne)

The next two terms have 197 and 1744 decimal digits, respectively.

a(n) gives the number of different ways in which a table of permutations for n objects can be arranged when one of them is fixed at the same place. Also: some of the first terms in this sequence belong to A010050. - R. J. Cano, Jan 23 2013

Cf. A000142, A000197, A010050.

PARI
```
a(n)=(n!-1)! \\ R. J. Cano, Jan 23 2013
```

a(n) = (n!-1)!.

a(n) = A000197(n)/n!.

Corrections made by Eric M. Schmidt, Jan 23 2013

A078670 Number of times n appears among the decimal digits of (n!)!.

Original entry on oeis.org

1, 1, 0, 4, 21, 169, 1504, 15755, 177333, 213789

Offset: 1

Jason Earls, Dec 16 2002

			a(4)=4 because 4 appears four times in (4!)! = 620448401733239439360000.

Cf. A000197.

Table[Count[Partition[IntegerDigits[(n!)!],IntegerLength[n],1], IntegerDigits[ n]],{n,10}] (* Harvey P. Dale, May 19 2014 *)
Table[SequenceCount[IntegerDigits[(n!)!],IntegerDigits[n]],{n,10}] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Aug 08 2016 *)

{mdcp(d,n)=local(a, c=0,L); L=length(Str(d)); if(L>1,a=2,a=1); while(n>0,if(n%(10^a)==d,n=floor(n/10); c++,n=floor(n/10); )); c } for(n=1,10,print1(mdcp(n,n!!)","))

Added a(9) and a(10). - Ryan Leonel (ryan.leonel(AT)gmail.com), Oct 03 2008

A102046 Smallest positive integer greater than a(n - 1) consistent with the condition that n is a member of the sequence if and only if a(n) is congruent to (n!)!.

Original entry on oeis.org

1, 1, 2, 6, 7, 8, 720, 721, 722, 723, 724, 726, 727, 728, 729, 780, 781, 782, 783, 784, 785, 786, 787, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799

Offset: 0

Michael Joseph Halm, Feb 12 2005

The sequence is related to the fake even and fake odd sequences and also the factorial and double factorial sequences, so seems in the short run linear but in the long run exponential.

			a(6) = 720 because (3!)! = 6! = 720

Cf. A080591, A080588, A000197, A000142.

a(a(n)) = (n!)!

The definition does not match the data. How was this sequence generated? - N. J. A. Sloane, Feb 21 2021

cp's OEIS Frontend

A336686 Decimal expansion of Sum_{k>=0} 1/(k!)!.

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A245087 Largest number such that 2^a(n) is a divisor of (n!)!.

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A301861 a(n) is the sum of the decimal digits of (n!)!.

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A348651 Number of ones in the binary expansion of (n!)!.

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A033187 a(n) = (n!)!/n!.

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A078670 Number of times n appears among the decimal digits of (n!)!.

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Mathematica

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Extensions

A102046 Smallest positive integer greater than a(n - 1) consistent with the condition that n is a member of the sequence if and only if a(n) is congruent to (n!)!.

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