cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

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

Examples

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

References

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

Links

Crossrefs

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.

Programs

  • Haskell
    a084558 n = a090529 (n + 1) - 1  -- Reinhard Zumkeller, Jan 05 2014
  • Maple
    0, seq(m$(m*m!),m=1..5); # Robert Israel, Apr 27 2015
  • Mathematica
    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 *)
  • PARI
    a(n)={my(m=0);while(n\=m++,);m-1} \\ R. J. Cano, Apr 09 2018
  • Python
    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

Formula

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

Extensions

Name clarified by Antti Karttunen, Apr 27 2015

A220658 Irregular table, where the n-th row consists of A084558(n)+1 copies of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 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, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2012

Keywords

Comments

Equally, for n>=1, each i in range [n!,(n+1)!-1] occurs n+1 times.
Used for computing A220659, A055089 and A060118: The n-th term a(n) tells which permutation (counted from the start, zero-based) of A055089 or A060117/A060118 the n-th term in those sequence belongs to.

Examples

			Rows of this irregular table begin as:
0;
1, 1;
2, 2, 2;
3, 3, 3;
4, 4, 4;
5, 5, 5;
6, 6, 6, 6;
The terms A055089(3), A055089(4) and A055089(5) are 1,3,2. As a(3), a(4) and a(5) are all 2, we see that "132" is the second permutation in A055089-list, after the identity permutation "1", which has the index zero.

Links

Crossrefs

Cf. A220657, A220659.

A220659 Irregular table: row n (n >= 1) consists of numbers 0..A084558(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3
Offset: 0

Views

Author

Antti Karttunen, Dec 18 2012

Keywords

Comments

The term a(n) gives the position (zero-based, starting from the left hand end of permutation) of a corresponding permutation A055089 and A060117/A060118 from which the term A055089(n) or A060118(n) is to be picked.

Examples

			Rows of this irregular table begin as:
0;
0, 1;
0, 1, 2;
0, 1, 2;
0, 1, 2;
0, 1, 2;
0, 1, 2, 3;
The term 2 occurs four times in A084558, in positions 2, 3, 4 and 5. Thus rows 2, 3, 4 and 5 (zero-based) of this irregular table are all 0,1,2.

Links

Programs

Formula

a(n) = n - A220657(A220658(n)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.