A027868 -id:A027868 - cp's OEIS frontend

A118813 Primes of the form (2n)! - n! - 1.

3628679, 87178286159, 20922789847679, 265252859812191058635000805631999

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 23 2006

nonn

The term a(7) (2407 digits) is too large to include even in the b-file.
All the numbers end in 9.
For all n, (2n)! - n! - 1 has exactly A027868(n) trailing 9s. - Rick L. Shepherd, Feb 16 2014
Generated by n = 5, 7, 8, 15, 103, 179, .... - R. J. Mathar, Sep 20 2012 (now A238011)

			For n=5, (2*5)! - 5! - 1 = 3628800 - 120 - 1 = 3628679, which is prime.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 159.

Rick L. Shepherd, Table of n, a(n) for n = 1..6

Cf. A238011 (corresponding n), A118812, A237443.

PFACT:=proc(N) local i,r; for i from 1 by 1 to N do r:=(2*i)!-i!-1; if isprime(r) then print(i); fi; od; end: PFACT(100);

A137581 Number of inner zeros in decimal representation of n!.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Jan 27 2008

Robert G. Wilson v, Table of n, a(n) for n = 0..10000.
Index entries for sequences related to factorial numbers.

Cf. A000142, A027869, A027868, A137582.

Cf. A004154, A055641.

a137581 = a055641 . a004154  -- Reinhard Zumkeller, Apr 01 2015

A137581:= proc(n) uses StringTools;
CountCharacterOccurrences(TrimRight(SubstituteAll(
    convert(n!,string),"0"," "))," ")
end proc; # Robert Israel, May 07 2012

f[n_] := f[n] = Block[{a = n!}, While[Mod[a, 10] == 0, a = a/10]; Count[ IntegerDigits@a, 0]]; Table[f@n, {n, 0, 98}] (* Robert G. Wilson v, Jan 28 2008 *)
Table[DigitCount[n!/10^IntegerExponent[n!,10],10,0],{n,0,100}] (* Harvey P. Dale, Apr 10 2023 *)

a(n) = A027869(n) - A027868(n);
a(A137582(n)) = 0.
a(n) = A055641(A004154(n)). - Reinhard Zumkeller, Apr 01 2015

A173345 Number of trailing zeros of the superfactorial of n (A000178).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 123, 131, 139, 147, 155, 164, 173, 182, 191, 200, 210, 220, 230, 240, 250, 262, 274, 286, 298, 310, 323, 336, 349, 362, 375, 389, 403, 417

Offset: 1

José María Grau Ribas, Feb 16 2010

Andrew Howroyd, Table of n, a(n) for n = 1..1000
A. M. Oller-Marcen and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011), Article 11.6.8.
Eric Weisstein's World of Mathematics, Superfactorial.

Cf. A000178, A027868.

a[n_] := Sum[Z10[i], {i, n}]; Z10[n_]:= Floor[Sum[Floor[n/5^i], {i, 1, Floor[Log[5, n]]}]]; Join[{0},Table[a[n], {n, 2, 200}]]
a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1];IntegerExponent[Table[a[n], {n, 1, 100}]] (* Stefano Spezia, Jan 26 2023 *)

a(n)=my(t=0);sum(k=5,n,t+=valuation(k,5)) \\ Charles R Greathouse IV, Jun 10 2011

a(n) = Sum_{k=1..n} A027868(k). - Charles R Greathouse IV, Jun 10 2011

A181576 Primes whose factorials end with a prime number of trailing 0's.

Original entry on oeis.org

11, 13, 17, 19, 31, 59, 83, 127, 151, 173, 179, 197, 199, 223, 293, 367, 397, 421, 439, 449, 461, 463, 557, 569, 607, 617, 619, 631, 659, 733, 773, 797, 853, 919, 941, 967, 1013, 1039, 1061, 1063, 1087, 1097, 1123, 1181, 1259, 1399, 1423, 1447, 1543, 1567

Offset: 1

Lekraj Beedassy, Nov 02 2010

For the corresponding prime number of trailing end 0's, see A181577.

			The factorial 2! = 2 ends with 0 zeros, so the prime 2 is not in the sequence because 0 is not a prime.
The factorial 5! = 120 ends with 1 zero, so the prime 5 is not in the sequence because 1 is not a prime.
The factorial 11! = 39916800 ends with 2 zeros, so the prime 11 is in the sequence because 2 is a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027868, A039716, A181577.

Select[ Prime@ Range@ 250, PrimeQ@ IntegerExponent[ #! ] &] (* Robert G. Wilson v, Nov 06 2010 *)

is(p) = isprime(p) && isprime((p - sumdigits(p, 5))/4); \\ Amiram Eldar, May 03 2024

A027868(a(n)) = A181577(n). - Amiram Eldar, May 03 2024

More terms from Robert G. Wilson v, Nov 06 2010

A181577 Prime number of trailing end 0's associated with p! where p = A181576(n).

Original entry on oeis.org

2, 2, 3, 3, 7, 13, 19, 31, 37, 41, 43, 47, 47, 53, 71, 89, 97, 103, 107, 109, 113, 113, 137, 139, 149, 151, 151, 157, 163, 181, 191, 197, 211, 227, 233, 239, 251, 257, 263, 263, 269, 271, 277, 293, 313, 347, 353, 359, 383, 389, 401, 421, 421, 433, 443, 463, 499

Offset: 1

Lekraj Beedassy, Nov 02 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027868, A181575, A181576.

Select[ IntegerExponent[ #! ] & /@ Prime@ Range@ 310, PrimeQ] (* Robert G. Wilson v, Nov 06 2010 *)

lista(pmax) = {my(tz); forprime(p = 1, pmax, tz = (p - sumdigits(p, 5))/4; if(isprime(tz), print1(tz, ", ")));} \\ Amiram Eldar, May 03 2024

a(n) = A027868(A181576(n)). - Amiram Eldar, May 03 2024

More terms from Robert G. Wilson v, Nov 06 2010

A181579 Smallest number m such that m! ends in exactly n trailing 0's (or 0 if no such m exists).

Original entry on oeis.org

0, 5, 10, 15, 20, 0, 25, 30, 35, 40, 45, 0, 50, 55, 60, 65, 70, 0, 75, 80, 85, 90, 95, 0, 100, 105, 110, 115, 120, 0, 0, 125, 130, 135, 140, 145, 0, 150, 155, 160, 165, 170, 0, 175, 180, 185, 190, 195, 0, 200, 205, 210, 215, 220, 0, 225, 230, 235, 240, 245, 0, 0, 250

Offset: 0

Lekraj Beedassy, Nov 02 2010

nonn

The sequence locates the first occurrence of n in A027868.

Entries are zero if n is an element of A000966.

Cf. A181578

a(n) = 5*A181578(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

A080066 First differences of A000966 (number of zeros that n! will never end in).

Original entry on oeis.org

6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 1, 6, 6, 6, 6, 6, 1, 6, 6, 6, 6, 6, 1, 6

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 24 2003

All terms in this sequence are either 1 or 6. - Charles R Greathouse IV, Jan 30 2023

The positions of 1 in this sequence are A000966, the sequence from which the differences were obtained. - Paul C Abbott, May 12 2025

			E.g. A000966(5)=29, A000966(4)=23, so a(4)=29-23=6.

Cf. A000966.

Cf. A000142, A027868.

a(n) = A000966(n+1) - A000966(n).

More terms from Sascha Kurz, Jan 27 2003

A116898 Numbers k such that k! is turned into a prime number by changing its trailing 0's into 1's.

Original entry on oeis.org

2, 10, 34, 499, 1746

Offset: 1

Giovanni Resta, Mar 07 2006

Also numbers k such that n! + R(Z(k)) is prime, where R(t) = (10^t - 1)/9 is the repunit with t digits (A002275) and Z(m) = Sum_{j>=1} floor(m/5^j) is the number of trailing zeros of m! (A027868). The (probable) prime corresponding to a(5)=1746 has 4905 digits. Next term must be greater than 4000.

Next term must be greater than 20000. - Michael S. Branicky, Nov 14 2024

			10 is a term, since 10! = 3628800 and 3628811 is prime.

Cf. A000142, A002275, A027868.

q:= n-> (f-> isprime(f+(10^padic[ordp](f, 10)-1)/9))(n!):
select(q, [$1..500])[];  # Alois P. Heinz, Feb 10 2021

tz1Q[n_]:=Module[{idn=Split[IntegerDigits[n!]]},PrimeQ[ FromDigits[ Flatten[ Join[ Most[ idn], Last[idn]/.(0->1)]]]]]; Select[ Range[ 1800],tz1Q] (* Harvey P. Dale, Oct 01 2015 *)

from sympy import isprime
from math import factorial
def ok(n):
  s, zeros = str(factorial(n)), 0
  while s[-1] == '0': s = s[:-1]; zeros += 1
  return isprime(int(s + '1'*zeros))
print([m for m in range(500) if ok(m)]) # Michael S. Branicky, Feb 10 2021

A135710 Positive integers b such that more than one prime factor p of b attains the maximum of (p-1)*v_p(b) where v_p(b) is the valuation of b at p.

Original entry on oeis.org

12, 45, 80, 90, 144, 180, 189, 240, 360, 378, 448, 637, 720, 756, 945, 1274, 1344, 1512, 1625, 1728, 1890, 1911, 2025, 2240, 2548, 2673, 3024, 3185, 3250, 3780, 3822, 4032, 4050, 4875, 5096, 5346, 5733, 6048, 6125, 6370, 6400, 6500, 6517, 6720, 7007, 7560, 7644

Offset: 1

Noam D. Elkies, Nov 25 2007

Given b, the number of trailing zeros at the end of the base-b representation of x! is asymptotic to x/M where M is the maximum over p|b of (p-1)*v_p(b).
Usually only one prime p attains the maximum and then the number is v_p(x!)/v_p(b) for all but finitely many x.
But for b=12,45,80,90,..., at least two v_p(x!) must be computed. For example: if b=12 then for x=2006 there are 998 trailing zeros due to v_3 but for x=2007 there are 999 due to v_2.

			For b=90 we have (p-1)*v_p(b) = 1, 4, 4 for p = 2, 3, 5 respectively so the maximum of 4 is attained twice (p=3 and p=5).

Eryk LIPKA, Automaticity of the sequence of the last nonzero digits of n! in a fixed base, Journal de Théorie des Nombres de Bordeaux 31 (2019), 283-291. [See Theorem 3.7 on page 290, and consider the complementary sequence.] - Jean-Paul Allouche and Don Reble, Oct 22 2020.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
J.-P. Allouche, J. Shallit, and R. Yassawi, How to prove that a sequence is not automatic, arXiv:2104.13072 [math.NT], 2021.

Cf. A027868, A011371, A054861.

a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 1, a(n-1)) while (s-> nops(s)<2 or (l->
      l[-2] (p-1)*padic[ordp](k, p),
       s))))([numtheory[factorset](k)[]]) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 23 2020

F[n_] := Module[{f, p, v, vmax}, f = FactorInteger[n]; p = f[[All, 1]]; v = Table[ f[[i, 2]]*(p[[i]]-1), {i, 1, Length[p]}]; vmax = Max[v]; Sum[Boole[v[[i]] == vmax], {i, 1, Length[v]}]]; Reap[For[n = 1, n <= 6400, n++, If[F[n] > 1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 09 2014, translated from PARI *)

{ F(n, f,p,v,vmax)= f=factor(n); p=f[,1]; v=vector(length(p),i,f[i,2]*(p[i]-1)); vmax=vecmax(v); sum(i=1,length(v),v[i]==vmax) } for(n=1,6400,if(F(n)>1,print(n)))

cp's OEIS Frontend

A118813 Primes of the form (2n)! - n! - 1.

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A137581 Number of inner zeros in decimal representation of n!.

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A173345 Number of trailing zeros of the superfactorial of n (A000178).

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A181576 Primes whose factorials end with a prime number of trailing 0's.

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A181577 Prime number of trailing end 0's associated with p! where p = A181576(n).

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A181579 Smallest number m such that m! ends in exactly n trailing 0's (or 0 if no such m exists).

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

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