A084557 -id:A084557 - 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

A030298 List of permutations of 1,2,3,...,n for n=1,2,3,..., in lexicographic order.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 1, 2, 3, 2, 1, 1, 2, 3, 4, 1, 2, 4, 3, 1, 3, 2, 4, 1, 3, 4, 2, 1, 4, 2, 3, 1, 4, 3, 2, 2, 1, 3, 4, 2, 1, 4, 3, 2, 3, 1, 4, 2, 3, 4, 1, 2, 4, 1, 3, 2, 4, 3, 1, 3, 1, 2, 4, 3, 1, 4, 2, 3, 2, 1, 4, 3, 2, 4, 1, 3, 4, 1, 2, 3, 4, 2, 1, 4, 1, 2, 3, 4, 1, 3, 2, 4, 2

Offset: 1

Clark Kimberling

Contains every finite sequence of distinct numbers, infinitely many times.

			The permutations can be written as
  1,
  12, 21,
  123, 132, 213, 231, 312, 321, etc.
Write them in order and insert commas.

Reinhard Zumkeller, Rows n=1..7 of triangle, flattened
Daniel Forgues, Tilman Piesk, et al., Orderings, OEIS Wiki.
Antti Karttunen, Ranking and unranking functions, OEIS Wiki.
Index entries for sequences related to permutations

A030299 gives the initial portion of these same permutations as decimally encoded numbers.
Cf. A098280, A098281, A030299, A170942.
Cf. A001563 (row lengths), A001286 (row sums).
Cf. A030496 for another ordering.
The same information is essentially given in A055089, but in more compact way, by skipping those permutations which start with a fixed element (cf. A220696).
A220660(n) tells the zero-based rank r of the n-th permutation in this sequence, among all finite permutations of the same size.
A220663(n) tells the zero-based position (from the left) of that a(n) in that permutation of rank r.
A084557(n) tells that the n-th term a(n) belongs to the a(n):th lexicographically ordered permutation from the start (its "global rank").
A220660(A084557(n)) tells the "local rank" of the permutation (amongst the permutations of the same size) to which the n-th term a(n) belongs.
(A130664(n),A084555(n)) = (1,1),(2,3),(4,5),(6,8),(9,11),(12,14),... gives the starting and ending offsets of the n-th permutation in this list.

import Data.List (permutations, sort)
a030298 n k = a030298_tabf !! (n-1) (k-1)
a030298_row = concat . sort . permutations . enumFromTo 1
a030298_tabf = map a030298_row [1..]
-- Reinhard Zumkeller, Mar 29 2012
(MIT/GNU Scheme, with Antti Karttunen's intseq-library):
;; Note that in Scheme, vector indexing is zero-based.
;; Requires also A055089permvec from A055089.
(define (A030298 n) (vector-ref (A030298permvec (A084556 (A084557 n)) (A220660 (A084557 n))) (A220663 n)))
(define (A030298permvec size rank) (vector-reverse (vector1invert (A055089permvec size rank))))
(define (vector1invert vec) (make-initialized-vector (vector-length vec) (lambda (i) (1+ (- (vector-length vec) (vector-ref vec i))))))
(define (vector-reverse vec) (make-initialized-vector (vector-length vec) (lambda (i) (vector-ref vec (- (vector-length vec) i 1)))))

f[n_] := Permutations[Range@ n, {n}]; Array[f, 4] // Flatten (* Robert G. Wilson v, Dec 18 2012 *)

from itertools import permutations, count, chain, islice
def A030298_gen(): # generator of terms
    return chain.from_iterable(p for l in count(2) for p in permutations(range(1,l)))
A030298_list = list(islice(A030298_gen(),30)) # Chai Wah Wu, Mar 21 2022

Start with 1, then 12 and 21, then the 6 permutations of 123 in lexical order: 123, 132, 213, 231, 312, 321 and so on.

Entry revised by N. J. A. Sloane, Feb 02 2006

Keyword tabf added by Reinhard Zumkeller, Mar 29 2012

A084556 n occurs n! times.

Original entry on oeis.org

0, 1, 2, 2, 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, 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, 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, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

Also minimum i such that A007489(i) >= n.

For n>=1, a(n) gives the length of the n-th permutation in the sequences like A030298 & A030496.

A. Karttunen, Table of n, a(n) for n = 0..46234
Index entries for sequences related to permutations

First differences of A084555. Used to compute A084557. Differs from A084506 first time at the 130th term, where A084506(130) = 6, while A084556(130) = 5. Cf. also A002024, A072643, A072649, A090529.

Flatten[ Table[#, {#!}] & /@ Range[0, 5]]

A084500 a(0)=0, after which each n occurs A084506(n) times.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 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, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

Also minimum i such that A084505(i) >= n. Tells that the n-th throw (n>=1) in A084501 belongs to the a(n)-th lexicographical solution A084502(a(n)).

Differs from A084557 first time at the 605th term, where A084500(605) = 130, while A084557(605) = 131. Cf. A084510, A084520.

A258198 a(n) = largest k for which A001563(k) = k*k! <= n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 23 2015

nonn

Number of nonzero terms of A001563 <= n.

Each n occurs A001564(n) times.

Antti Karttunen, Table of n, a(n) for n = 0..4320

Cf. A001563, A001564, A084556, A084557, A256450, A258199.

(define (A258198 n) (let loop ((k 1) (f 1)) (if (> (* k f) n) (- k 1) (loop (+ k 1) (* (+ k 1) f)))))

A220662 Irregular table: row n (n>=1) consists of A084556(n) copies of A130664(n).

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 6, 6, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 21, 21, 21, 24, 24, 24, 24, 28, 28, 28, 28, 32, 32, 32, 32, 36, 36, 36, 36, 40, 40, 40, 40, 44, 44, 44, 44, 48, 48, 48, 48, 52, 52, 52, 52, 56, 56, 56, 56, 60, 60, 60, 60, 64, 64, 64, 64, 68

Offset: 1

Antti Karttunen, Dec 18 2012

The term a(n) gives the position of the first term in A030298 which belongs to the same permutation as A030298(n) itself.

			Rows of this irregular table begin as:
1;
2, 2;
4, 4;
6, 6, 6;
9, 9, 9;

A. Karttunen, Rows 1..998 of the irregular table, flattened.

Cf. A220663, A220660, A030298.

a(n) = A130664(A084557(n)) = A084555(A084557(n)-1)+1.

a(A130664(n)) = A130664(n).

cp's OEIS Frontend

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

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A030298 List of permutations of 1,2,3,...,n for n=1,2,3,..., in lexicographic order.

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A084556 n occurs n! times.

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A084500 a(0)=0, after which each n occurs A084506(n) times.

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A258198 a(n) = largest k for which A001563(k) = k*k! <= n.

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A220662 Irregular table: row n (n>=1) consists of A084556(n) copies of A130664(n).

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