A111095 -id:A111095 - cp's OEIS frontend

A084558 a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.

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

Offset: 0

Antti Karttunen, Jun 23 2003

For n >= 1, a(n) = the number of significant digits in n's factorial base representation (A007623).
After zero, which occurs once, each n occurs A001563(n) times.
Number of iterations (...(f_4(f_3(f_2(n))))...) such that the result is < 1, where f_j(x):=x/j. - Hieronymus Fischer, Apr 30 2012
For n > 0: a(n) = length of row n in table A108731. - Reinhard Zumkeller, Jan 05 2014

			a(4) = 2 because 2! <= 4 < 3!.

F. Smarandache, "f-Inferior and f-Superior Functions - Generalization of Floor Functions", Arizona State University, Special Collections.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Yi Yuan and Zhang Wenpeng, On the Mean Value of the Analogue of Smarandache Function, Smarandache Notions J., Vol. 15.

A dual to A090529.
Cf. A084555, A084556, A084557.
Cf. A001069, A010096.
Cf. A000142, A001563, A055089, A060130, A111095, A211664, A211670, A108731, A212598, A220656, A220657, A220658, A220659, A231716, A235224, A257510.

a084558 n = a090529 (n + 1) - 1  -- Reinhard Zumkeller, Jan 05 2014

0, seq(m$(m*m!),m=1..5); # Robert Israel, Apr 27 2015

Table[m = 1; While[m! <= n, m++]; m - 1, {n, 0, 104}] (* Jayanta Basu, May 24 2013 *)
Table[Floor[Last[Reduce[x! == n && x > 0, x]]], {n, 120}] (* Eric W. Weisstein, Sep 13 2024 *)

a(n)={my(m=0);while(n\=m++,);m-1} \\ R. J. Cano, Apr 09 2018

def A084558(n):
  i=1
  while n: i+=1; n//=i
  return(i-1)
print(list(map(A084558,range(101)))) # Natalia L. Skirrow, May 28 2023

From Hieronymus Fischer, Apr 30 2012: (Start)
a(n!) = a((n-1)!)+1, for n>1.
G.f.: 1/(1-x)*Sum_{k>=1} x^(k!).
The explicit first terms of the g.f. are: (x+x^2+x^6+x^24+x^120+x^720...)/(1-x).
(End)
Other identities:
For all n >= 0, a(n) = A090529(n+1) - 1. - Reinhard Zumkeller, Jan 05 2014
For all n >= 1, a(n) = A060130(n) + A257510(n). - Antti Karttunen, Apr 27 2015
a(n) ~ log(n^2/(2*Pi)) / (2*LambertW(log(n^2/(2*Pi))/(2*exp(1)))) - 1/2. - Vaclav Kotesovec, Aug 22 2025

Name clarified by Antti Karttunen, Apr 27 2015

A108911 Difference between n and the sum of the factorials of its digits.

Original entry on oeis.org

0, 0, -3, -20, -115, -714, -5033, -40312, -362871, 8, 9, 9, 6, -11, -106, -705, -5024, -40303, -362862, 17, 18, 18, 15, -2, -97, -696, -5015, -40294, -362853, 23, 24, 24, 21, 4, -91, -690, -5009, -40288, -362847, 15, 16, 16, 13, -4, -99, -698, -5017, -40296, -362855, -71, -70, -70, -73

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 18 2005

Null values are at n = 1, 2, 145, 40585 (A014080). Twin values are at n = 1, 2; 11, 12; 21, 22; ... 10*i + 1, 10*i + 2. Not in sequence: 7, 10, 14, ... Nice polar diagrams repeating themselves with normalized angle to 9! and radius = a(n).

The sequence can be seen as the difference between the natural numbers in the decimal system (n_dec = N0*(10^0) + N1*(10^1) + N2*(10^2)...) and their values in a non-positional number system based on the factorials of the digits (n_fact = N0*(N0 - 1)! + N1*(N1 - 1)! + N2*(N2 - 1)! ...). See also A111095. Note that a(np) - a(n) is congruent to 0 mod 9 if n and np are different for the permutation of the digits. Example (a(5971) - a(1957))/9 = 446. The property can be easily derived by remembering that np - n is congruent to 0 mod 9. - Giorgio Balzarotti, Oct 15 2005

			For n = 35, a(35) = -91 because 35 - (3! + 5!) = 35 - (6 + 120) = -91.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A005096, A014080, A061602.

[n-&+[Factorial(d): d in Intseq(n)]: n in [1..60]]; // Bruno Berselli, Oct 25 2018

a:= n-> n-add(i!, i=convert(n, base, 10)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 24 2018

f[n_] := n - Plus @@ Factorial /@ IntegerDigits[n]; Table[f[n], {n, 53}] (* Ray Chandler, Jul 24 2005 *)

a(n) = my(d = digits(n)); n - sum(i=1, #d, d[i]!); \\ Michel Marcus, Apr 21 2014

a(n) = n - (N0! + N1! + N2! + ...) if n = N0*10^0 + N1*10^1 + N2*10^2 ...

a(n) = n - A061602(n). - Michel Marcus, Apr 21 2014

Extended by Ray Chandler, Jul 24 2005

A181521 Representation of n = sum_k b_k*(k!!) in the double-factorial base by some b_k-fold concatenation of the indices k.

Original entry on oeis.org

1, 2, 3, 13, 23, 33, 133, 4, 14, 24, 34, 134, 234, 334, 5, 15, 25, 35, 135, 235, 335, 1335, 45, 145, 245, 345, 1345, 2345, 3345, 55, 155, 255, 355, 1355, 2355, 3355, 13355, 455, 1455, 2455, 3455, 13455, 23455, 33455, 555, 1555, 2555, 6, 16, 26, 36, 136, 236, 336

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Oct 26 2010

The encoding of n is similar to A111095 but uses a double-factorial base A006882 to define the expansion coefficients.
The expansion coefficients b_k in n = sum_{k>=1} b_k * A006882(k) are defined "greedily" by taking the largest A006882(k) which is <=n, choosing b_k as large as possible such that b_k*A006882(k) remains <=n, subtracing b_k*A006882(k) from n to define a remainder, and recursively slicing the remainder to generate b_{k-1}, then b_{k-2} etc until the remainder reduces to zero. This produces the b_k for each n equivalent to A019513(n).
This representation A019513 is then scanned from the least to the most-significant b_k, i.e., along increasing k, and for each nonzero b_k, b_k copies of k are appended to a string representation -- starting from an empty string. This final representation is interpreted as a base-10 number a(n).

			a(39) = 1455 because 1!!+4!!+5!!+5!! = 1+8+15+15 = 39

Cf. A111095

dblfactfloor := proc(n) local j ; for j from 1 do if doublefactorial(j) > n then return j-1 ; end if; end do: end proc:
dblfbase := proc(n) local nshf,L,f; nshf := n ; L := [] ; while nshf > 0 do f := dblfactfloor(nshf) ; L := [f,op(L)] ; nshf := nshf-doublefactorial(f) ; end do: L ; end proc:
read("transforms") ; A181521 := proc(n) digcatL(dblfbase(n)) ; end proc:
seq(A181521(n),n=1..70) ; # R. J. Mathar, Dec 06 2010

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A084558 a(0) = 0; for n >= 1: a(n) = largest m such that n >= m!.

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A108911 Difference between n and the sum of the factorials of its digits.

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A181521 Representation of n = sum_k b_k*(k!!) in the double-factorial base by some b_k-fold concatenation of the indices k.

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Maple