A027868 -id:A027868 - cp's OEIS frontend

A090619 Highest power of 12 dividing n!.

0, 0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 5, 5, 6, 6, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 15, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 22, 22, 22, 23, 23, 23, 25, 25, 26, 26, 27, 27, 28, 28, 28, 28, 30, 30, 31, 31, 31, 32, 32, 32, 34, 34, 34, 35, 35

Offset: 0

Henry Bottomley, Dec 06 2003

nonn

Most sequences of the form "highest power of k dividing n!" essentially depend on one of the primes or prime powers dividing k. But in this case, the sequences with k=3 (A054861) and k=4 (A090616) are both close to n/2 and vary in which one is lower for different values of n.

a(2^n) = A090616(2^n) and a(3^n-1) = A090616(3^n-1) while a(2^n-1) = A054861(2^n-1) and a(3^n) = A054861(3^n). - Robert Israel, Mar 25 2018

			a(6)=2 since 6!=720=12^2*5.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A011371, A054861, A090616, A027868, A054896, A090617, A090618.

f2:= n -> n - convert(convert(n,base,2),`+`):
f3:= n -> (n - convert(convert(n,base,3),`+`))/2:
f:= n -> min(f3(n), floor(f2(n)/2)):
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Mar 23 2018

Table[IntegerExponent[n!, 12], {n, 0, 100}] (* Jean-François Alcover, Mar 26 2018 *)

a(n) = valuation(n!, 12); \\ Michel Marcus, Mar 24 2018

a(n) =A090622(n, 12) =min(A054861(n), A090616(n)). Close to n/2, indeed for n>3: n/2-log3(n+1) <= a(n) < n/2.

A102684 Number of times the digit 9 appears in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20

Offset: 0

N. J. A. Sloane, Feb 03 2005

This is the total number of digits = 9 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Partial sums of A102683.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=9 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..105); # Emeric Deutsch, Feb 23 2005

Accumulate[DigitCount[Range[0,100],10,9]] (* Harvey P. Dale, Mar 30 2018 *)

a(n) = sum(k=0, n, #select(x->(x==9), digits(k))); \\ Michel Marcus, Oct 03 2023

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/10)*(2n + 2 - (4/5 + floor(n/10^j + 1/10))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
a(n) = (n+1)*A102683(n) + (1/2)*Sum_{j=1..m+1} ((-4/5*floor(n/10^j + 1/10) + floor(n/10^j))*10^j - (floor(n/10^j + 1/10)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = m*10^(m-1).
(this is total number of digits = 9 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(9*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

Definition revised by N. J. A. Sloane, Mar 30 2018

A243757 a(n) = Product_{i=1..n} A060904(i).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 5, 5, 5, 5, 25, 25, 25, 25, 25, 125, 125, 125, 125, 125, 625, 625, 625, 625, 625, 15625, 15625, 15625, 15625, 15625, 78125, 78125, 78125, 78125, 78125, 390625, 390625, 390625, 390625, 390625, 1953125, 1953125, 1953125, 1953125, 1953125, 9765625

Offset: 0

Tom Edgar, Jun 10 2014

nonn

This is the generalized factorial for A060904.
a(0) = 1 as it represents the empty product.
a(n) is the largest power of 5 that divides n!, or the order of a 5-Sylow subgroup of the symmetric group of degree n. - David Radcliffe, Sep 03 2021

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Cf. A027868, A060818, A060828, A060904, A242954.

a243757 n = a243757_list !! n
a243757_list = scanl (*) 1 a060904_list
-- Reinhard Zumkeller, Feb 04 2015

Table[Product[5^IntegerExponent[k, 5], {k, 1, n}], {n, 0, 20}] (* G. C. Greubel, Dec 24 2016 *)

a(n) = prod(k=1,n, 5^valuation(k,5)); \\ G. C. Greubel, Dec 24 2016

S=[0]+[5^valuation(i, 5) for i in [1..100]]
[prod(S[1:i+1]) for i in [0..99]]

a(n) = Product_{i=1..n} A060904(i).

a(n) = 5^(A027868(n)).

A104355 Number of trailing zeros in decimal representation of A104350(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 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, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Reinhard Zumkeller, Mar 06 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI link
Reinhard Zumkeller, Products of largest prime factors of numbers <= n.

Cf. A000079, A027868, A080193, A104350, A104356, A122840.

IntegerExponent[FoldList[Times, Array[FactorInteger[#][[-1, 1]] &, 100]], 10] (* Amiram Eldar, Apr 08 2024 *)

gpf(n) = {my(p = factor(n)[, 1]); p[#p];}
a(n) = valuation(prod(k = 2, n, gpf(k)), 10); \\ Amiram Eldar, Apr 08 2024

\\ See link. David A. Corneth, Apr 08 2024

a(A104356(n)) = n and a(m) < n for m < A104356(n);

a(n) = A122840(A104350(n)). - Reinhard Zumkeller, Mar 10 2013

A246839 Number of trailing zeros in A002109(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 5, 5, 5, 5, 15, 15, 15, 15, 15, 30, 30, 30, 30, 30, 50, 50, 50, 50, 50, 100, 100, 100, 100, 100, 130, 130, 130, 130, 130, 165, 165, 165, 165, 165, 205, 205, 205, 205, 205, 250, 250, 250, 250, 250, 350, 350, 350, 350, 350, 405, 405, 405, 405

Offset: 0

Chai Wah Wu, Sep 04 2014

Chai Wah Wu, Table of n, a(n) for n = 0..999

Cf. A002109, A027868, A112765, A122840, A246817, A249152.

(n=#;k=0;While[Mod[n,10]==0,n=n/10;k++];k)&/@Hyperfactorial@Range[0,60] (* Giorgos Kalogeropoulos, Sep 14 2021 *)

a(n) = sum(i=1, n, i*valuation(i, 5)); \\ Michel Marcus, Sep 14 2021

def a(n):
  s = 1
  for k in range(n+1):
    s *= k**k
  i = 1
  while not s % 10**i:
    i += 1
  return i-1
n = 1
while n < 100:
  print(a(n),end=', ')
  n += 1 # Derek Orr, Sep 04 2014

from sympy import multiplicity
A246839, p5 = [0,0,0,0,0], 0
for n in range(5,10**3,5):
    p5 += multiplicity(5,n)*n
    A246839.extend([p5]*5)
# Chai Wah Wu, Sep 05 2014

From Michel Marcus, Sep 14 2021: (Start)
a(n) = A122840(A002109(n)), but also,
a(n) = A112765(A002109(n)), see explanation in A002109; so
a(n) = Sum_{i=1..n} i*v_5(i), where v_5(i) = A112765(i) is the exponent of the highest power of 5 dividing i. After a similar formula in A249152. (End)
a(n) = A246817(floor(n/5)+1). - Jason Bard, Sep 06 2025

A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

Original entry on oeis.org

0, 1, 4, 13, 40, 186, 952, 5533, 38719, 346207, 3130816, 34444968, 382437431, 4637235152

Offset: 2

Antti Karttunen, Nov 06 2024

For 1! = 1, there are an infinite number of integers k for which A003415(k) = 1 (namely, all the primes), therefore the starting offset is 2.
Like with A351029, also here most of the solutions seem to be squarefree semiprimes, counted by A062311.
Terms a(12)..a(15) were obtained by summing the corresponding terms of A062311 and A377986.

Index entries for sequences related to factorial numbers

Cf. A000142, A002620, A003415, A062311, A099302, A377986.
Cf. A011371, A054861, A027868, A054896.
Cf. also A351029, A369239.

\\ Slow program, for computing just a few terms:
A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A376410(n) = { my(g=n!); sum(k=1,A002620(g),A003415(k)==g); };

A376410(n) = AntiDeriv(n!);
AntiDeriv(n,startvlen=1,solsfilename="") = { my(v = vector(startvlen,i,2), ip = #v, r, c=0); while(1, r = A003415vrl(v,n); if(0==r, ip--, if(r > 1, c++; if(solsfilename!="", write(solsfilename, r*factorback(v)))); ip = #v); if(0==ip, v = vector(1+#v,i,2); ip = #v; if(A003415vec(v) > n, return(c)), v[ip] = nextprime(1+v[ip]); for(i=1+ip, #v, v[i]=v[i-1]))); };
A003415vec(tv) = { my(n=factorback(tv), s=0, m=1, spf); for(i=1,#tv,spf = tv[i]; n /= spf; s += m*n; m *= spf); (s); }; \\ Compute Arithmetic derivative from the vector of primes.
A003415vrl(pv,lim) = { my(n=factorback(pv), x=lim-n, s=0, m=1, spf, u=n); for(i=1,#pv,spf = pv[i]; u /= spf; s += m*u; m *= spf); if(((x/s)

a(n) = A099302(A000142(n)).
a(n) = Sum_{k=1..A002620(n!)} [A003415(k) = n!], where [ ] is the Iverson bracket.
a(n) = A062311(n) + A377986(n).

A000976 Period of 1/n! in base 10.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 6, 6, 18, 18, 18, 54, 54, 378, 1134, 1134, 9072, 81648, 81648, 81648, 1714608, 18860688, 18860688, 56582064, 56582064, 735566832, 19860304464, 139022131248, 139022131248, 417066393744, 2085331968720, 2085331968720, 68815954967760

Offset: 1

Simon Plouffe

T. D. Noe, Table of n, a(n) for n = 1..200
Michael Penn, From the GOAT number theory book!, YouTube video, 2022.

Cf. A000142, A051626.

Join[{0, 0}, Table[num = n!/(2^IntegerExponent[n!, 2] * 5^IntegerExponent[n!, 5]); MultiplicativeOrder[10, num], {n, 3, 30}]] (* T. D. Noe, Jun 21 2012 *)

a(n) = if(n <= 2, return(0)); znorder(Mod(10,n!/2^val(n,2)/5^val(n,5)))
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Jan 11 2023

a(n) = k where k is the smallest integer >= 1 such that 10^k == 1 (mod n!/(2^A011371(n)*5^A027868(n))) where A011371(n) is the highest power of 2 dividing n! and A027868(n) is the largest k such that 5^k | n!. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004, corrected by David A. Corneth, Jan 11 2023
a(n) = order(10, n!/(2^s*5^t)) where 2^s is largest power of 2 dividing n! and 5^t is largest power of 5 dividing n!. - Sean A. Irvine, Sep 29 2011
a(n) = A051626(A000142(n)). - Michel Marcus, Jan 12 2023

One more term from Sean A. Irvine, Sep 28 2011

A008906 Number of digits in n! excluding final zeros.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 15, 16, 18, 19, 20, 20, 21, 23, 24, 25, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 39, 41, 43, 44, 46, 47, 48, 50, 52, 53, 53, 55, 56, 58, 60, 61, 62, 64, 66, 68, 68, 70, 72, 74, 76, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 92, 94

Offset: 0

Russ Cox

From Bernard Schott, Nov 19 2021: (Start)
a(n) < n iff 2 <= n <= 38 or n = 40;
a(n) = n iff n = 1, 39, 41;
a(n) > n iff n = 0 or n >= 42. (End)

Cf. A027868, A034886.

Array[IntegerLength[#!//.x_/;x~Mod~10==0:>x/10]&,77,0] (* Giorgos Kalogeropoulos, Nov 19 2021 *)

from math import factorial
def A008906(n): return len(str(factorial(n)).rstrip('0')) # Chai Wah Wu, Oct 24 2021

a(n) = A034886(n) - A027868(n). - Michel Marcus, Jun 24 2013

A031145 Factorials with a record number of zeros.

Original entry on oeis.org

1, 120, 5040, 479001600, 6402373705728000, 121645100408832000, 2432902008176640000, 1124000727777607680000, 15511210043330985984000000, 304888344611713860501504000000

Offset: 0

N. J. A. Sloane

			From _Alonso del Arte_, May 19 2017: (Start)
Note that 5040 has two zeros, even though only one of them is a trailing zero.
Although 3628800 has one more zero than 362880, it still has as many zeros as 5040, and for that reason it is not in this sequence.
Thus the next term after 5040 is 479001600, which has four zeros. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..59

Cf. A031144 (indices of factorials), A027868.

Function[s, Map[Position[s, #][[1, 1]] &, Union@ FoldList[Max, s]]! ]@ Table[DigitCount[n!, 10, 0], {n, 28}] (* Michael De Vlieger, May 20 2017 *)
DeleteDuplicates[Table[{n!,DigitCount[n!,10,0]},{n,50}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Dec 03 2023 *)

More terms from Erich Friedman.

Name clarified by Alonso del Arte, May 19 2017

A090621 Exponent of highest power of 16 dividing n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 0

Henry Bottomley, Dec 06 2003

nonn

			a(10)=2 since 10! = 3628800 = 16^2 * 14175.

Cf. A011371, A054861, A090616, A027868, A054896, A090617, A090618, A090619, A090620.

a(n) = valuation(n!, 16); \\ Michel Marcus, Jul 10 2022

a(n) = A090622(n, 16) = floor(A011371(n)/4) = floor(A090616(n)/2) = floor((floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...)/4). Almost n/4.

cp's OEIS Frontend

A090619 Highest power of 12 dividing n!.

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A102684 Number of times the digit 9 appears in the decimal representations of all integers from 0 to n.

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A243757 a(n) = Product_{i=1..n} A060904(i).

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A104355 Number of trailing zeros in decimal representation of A104350(n).

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A246839 Number of trailing zeros in A002109(n).

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A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

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A000976 Period of 1/n! in base 10.

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