A064039 -id:A064039 - cp's OEIS frontend

A007623 Integers written in factorial base.

0, 1, 10, 11, 20, 21, 100, 101, 110, 111, 120, 121, 200, 201, 210, 211, 220, 221, 300, 301, 310, 311, 320, 321, 1000, 1001, 1010, 1011, 1020, 1021, 1100, 1101, 1110, 1111, 1120, 1121, 1200, 1201, 1210, 1211, 1220, 1221, 1300, 1301, 1310, 1311, 1320, 1321, 2000, 2001, 2010

Offset: 0

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Places reading from right have values (1, 2, 6, 24, 120, ...) = factorials.
Also the reversed inversion vectors for the list of all finite permutations in reversed lexicographic order: A055089.
This concatenated representation is unsatisfactory for large n (above 36287999), when coefficients of 10 or greater start to appear. For these large numbers the representation given in A108731 is better. - N. J. A. Sloane, Jun 04 2012
For n < 10*10!-1, a(n) = concatenation of n-th row of triangle in A108731. - Reinhard Zumkeller, Jun 04 2012
a(n) = A049345(n) for n=0..23. - Reinhard Zumkeller, Jan 05 2014
For n = 36288000 = 10 * 10!, the digits in factorial base are {10, 0, 0, 0, 0, 0, 0, 0, 0, 0}. - Michael De Vlieger, Oct 11 2015, corrected and edited by M. F. Hasler, Nov 27 2018
The alt text in xkcd comic #2835 describes "Numbers larger than about 3.6 million" to be illegal. See links. - David Cleaver, Sep 30 2023

			a(47) = 1321 because 47 = 1*4! + 3*3! + 2*2! + 1*1!

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 2, p. 192.
F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House, 2000.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. F. Hasler (terms 0 .. 1000) & Antti Karttunen, Table of n, a(n) for n = 0..40320
Italo J. Dejter, A numeral system for the middle levels, arXiv:1012.0995 [math.CO], 2010-2015.
Italo J. Dejter, Dihedral-symmetry middle-levels problem via a Catalan system of numeration, preprint, 2015.
Italo J. Dejter, Ordering the Levels Lk and Lk+1 of B2k+1, preprint, 2015-2017.
P. Hecht, Post-Quantum Cryptography: S_381 Cyclic Subgroup of High Order, International Journal of Advanced Engineering Research and Science (IJAERS, 2017) Vol. 4, Issue 6, 78-86.
C. A. Laisant, Sur la numération factorielle, application aux permutations, Bulletin de la Société Mathématique de France, 16 (1888), p. 176-183.
Randall Munroe, Factorial Numbers, xkcd Web Comic #2835, Sep 29 2023.
Wikipedia, Factorial base
Index entries for sequences related to factorial base representation

Cf. A000142, A034968 (sum of digits), A060130 (number of nonzero digits), A099563 (the most significant digit).
Cf. also A055089, A055881, A060112, A060495. Permutation of A064039.
See index entry "factorial base representation" for many more related sequences.
See also primorial base A049345.

a007623 n | n <= 36287999 = read $ concatMap show (a108731_row n) :: Int
          | otherwise     = error "representation would be ambiguous"
-- Reinhard Zumkeller, Jun 04 2012
(Scheme, R6RS standard) (define (A007623 n) (let loop ((n n) (s 0) (p 1) (i 2)) (if (zero? n) s (let ((d (mod n i))) (loop (/ (- n d) i) (+ (* p d) s) (* 10 p) (+ 1 i)))))) ;; In older Schemes use modulo instead of mod. - Antti Karttunen, Feb 13 2016

a := n -> if nargs<2 then a(n,2) elif n

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++ ]; m = n; len = i; dList = Table[0, {len}]; Do[ currDigit = 0; While[m >= j!, m = m - j!; currDigit++ ]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; Table[FromDigits[factBaseIntDs[n]], {n, 0, 50}] (* Alonso del Arte, May 03 2006 *)
lim = 50; m = 1; While[Factorial@ m < lim, m++]; m; IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] & /@ Range@ lim (* Michael De Vlieger, Oct 11 2015, Version 10.2 *)

apply( a(n,p=2)=if(nM. F. Hasler, Mar 27 2007; minor edit Nov 26 2018

def a(n, p=2): return n if nIndranil Ghosh, Jun 19 2017, after PARI program
(APL, Dyalog dialect) A007623 ← {⍺←2 ⋄ ⍵<⍺: ⍵ ⋄ (⍺|⍵) + ((⍺+1) ∇ ⍺(⌊÷⍨)⍵)×10} ⍝ (code from Dani Adham) - Antti Karttunen, Feb 20 2024

More terms from R. K. Guy

A034968 Minimal number of factorials that add to n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7

Offset: 0

Erich Friedman

nonn

Equivalently, sum of digits when n is written in factorial base (A007623).
Equivalently, a(0)...a(n!-1) give the total number of inversions of the permutations of n elements in lexicographic order (the factorial numbers in rising base are the inversion tables of the permutations and their sum of digits give the total number of inversions, see example and the Fxtbook link). - Joerg Arndt, Jun 17 2011
Also minimum number of adjacent transpositions needed to produce each permutation in the list A055089, or number of swappings needed to bubble sort each such permutation. (See A055091 for the minimum number of any transpositions.)

			a(205) = a(1!*1 + 3!*2 + 4!*3 + 5!*1) = 1+2+3+1 = 7. [corrected by Shin-Fu Tsai, Mar 23 2021]
From _Joerg Arndt_, Jun 17 2011: (Start)
   n:    permutation   inv. table a(n)  cycles
   0:    [ 0 1 2 3 ]   [ 0 0 0 ]   0    (0) (1) (2) (3)
   1:    [ 0 1 3 2 ]   [ 0 0 1 ]   1    (0) (1) (2, 3)
   2:    [ 0 2 1 3 ]   [ 0 1 0 ]   1    (0) (1, 2) (3)
   3:    [ 0 2 3 1 ]   [ 0 1 1 ]   2    (0) (1, 2, 3)
   4:    [ 0 3 1 2 ]   [ 0 2 0 ]   2    (0) (1, 3, 2)
   5:    [ 0 3 2 1 ]   [ 0 2 1 ]   3    (0) (1, 3) (2)
   6:    [ 1 0 2 3 ]   [ 1 0 0 ]   1    (0, 1) (2) (3)
   7:    [ 1 0 3 2 ]   [ 1 0 1 ]   2    (0, 1) (2, 3)
   8:    [ 1 2 0 3 ]   [ 1 1 0 ]   2    (0, 1, 2) (3)
   9:    [ 1 2 3 0 ]   [ 1 1 1 ]   3    (0, 1, 2, 3)
  10:    [ 1 3 0 2 ]   [ 1 2 0 ]   3    (0, 1, 3, 2)
  11:    [ 1 3 2 0 ]   [ 1 2 1 ]   4    (0, 1, 3) (2)
  12:    [ 2 0 1 3 ]   [ 2 0 0 ]   2    (0, 2, 1) (3)
  13:    [ 2 0 3 1 ]   [ 2 0 1 ]   3    (0, 2, 3, 1)
  14:    [ 2 1 0 3 ]   [ 2 1 0 ]   3    (0, 2) (1) (3)
  15:    [ 2 1 3 0 ]   [ 2 1 1 ]   4    (0, 2, 3) (1)
  16:    [ 2 3 0 1 ]   [ 2 2 0 ]   4    (0, 2) (1, 3)
  17:    [ 2 3 1 0 ]   [ 2 2 1 ]   5    (0, 2, 1, 3)
  18:    [ 3 0 1 2 ]   [ 3 0 0 ]   3    (0, 3, 2, 1)
  19:    [ 3 0 2 1 ]   [ 3 0 1 ]   4    (0, 3, 1) (2)
  20:    [ 3 1 0 2 ]   [ 3 1 0 ]   4    (0, 3, 2) (1)
  21:    [ 3 1 2 0 ]   [ 3 1 1 ]   5    (0, 3) (1) (2)
  22:    [ 3 2 0 1 ]   [ 3 2 0 ]   5    (0, 3, 1, 2)
  23:    [ 3 2 1 0 ]   [ 3 2 1 ]   6    (0, 3) (1, 2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
Joerg Arndt, Matters Computational (The Fxtbook), fig.10-1.B on p.234.
Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.
FindStat - Combinatorial Statistic Finder, The number of inversions of a permutation
Index entries for sequences related to factorial base representation

Cf. A368342 (partial sums), A001809 (sums of n! terms).
Cf. A000142, A007489, A007623, A033312, A055091, A139365.
Cf. A000217, A001222, A056019, A060130, A099563, A225901, A257684, A257687, A257694, A276076, A276146.
Cf. A227148 (positions of even terms), A227149 (of odd terms).
Cf. also A219650, A219651, A219666, A230423.
Differs from analogous A276150 for the first time at n=24.
Positions of records are A200748.

[seq(convert(fac_base(j),`+`),j=0..119)]; # fac_base and PermRevLexUnrank given in A055089. Perm2InversionVector in A064039
Or alternatively: [seq(convert(Perm2InversionVector(PermRevLexUnrank(j)),`+`),j=0..119)];
# third Maple program:
b:= proc(n, i) local q;
      `if`(n=0, 0, b(irem(n, i!, 'q'), i-1)+q)
    end:
a:= proc(n) local k;
      for k while k!Alois P. Heinz, Nov 15 2012

a[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Nov 06 2013, after Benoit Cloitre *)

a(n)=local(k,r);k=2;r=0;while(n>0,r+=n%k;n\=k;k++);r \\ Franklin T. Adams-Watters, May 13 2009

def a(n):
    k=2
    r=0
    while n>0:
        r+=n%k
        n=n//k
        k+=1
    return r
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 19 2017, after PARI program

def A034968(n, p=2): return n if nA034968(n//p, p+1) + n%p
print([A034968(n) for n in range(106)]) # Michael S. Branicky, Oct 27 2024

(define (A034968 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (quotient n i) (+ 1 i) (+ s (remainder n i)))))))
;; Antti Karttunen, Aug 29 2016

a(n) = n - Sum_{i>=2} (i-1)*floor(n/i!). - Benoit Cloitre, Aug 26 2003
G.f.: 1/(1-x)*Sum_{k>0} (Sum_{i=1..k} i*x^(i*k!))/(Sum_{i=0..k} x^(i*k!)). - Franklin T. Adams-Watters, May 13 2009
From Antti Karttunen, Aug 29 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A099563(n) + a(A257687(n)).
a(0) = 0; for n >= 1, a(n) = A060130(n) + a(A257684(n)).
Other identities. For all n >= 0:
a(n) = A001222(A276076(n)).
a(n) = A276146(A225901(n)).
a(A000142(n)) = 1, a(A007489(n)) = n, a(A033312(n+1)) = A000217(n).
a(A056019(n)) = a(n).
A219651(n) = n - a(n).
(End)

Additional comments from Antti Karttunen, Aug 23 2001

A060117 A list of all finite permutations in "PermUnrank3R" ordering. (Inverses of the permutations of A060118.)

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 02 2001

PermUnrank3R and PermUnrank3L are slight modifications of unrank2 algorithm presented in Myrvold-Ruskey article.

			In this table each row consists of A001563[n] permutations of (n+1) terms; i.e., we have (1/) 2,1/ 1,3,2; 3,1,2; 3,2,1; 2,3,1/ 1,2,4,3; 2,1,4,3;
Append to each an infinite number of fixed terms and we get a list of rearrangements of natural numbers, but with only a finite number of terms permuted:
1/2,3,4,5,6,7,8,9,...
2,1/3,4,5,6,7,8,9,...
1,3,2/4,5,6,7,8,9,...
3,1,2/4,5,6,7,8,9,...
3,2,1/4,5,6,7,8,9,...
2,3,1/4,5,6,7,8,9,...
1,2,4,3/5,6,7,8,9,...
2,1,4,3/5,6,7,8,9,...

W. Myrvold and F. Ruskey, Ranking and Unranking Permutations in Linear Time, Inform. Process. Lett. 79 (2001), no. 6, 281-284.

A060119 = Positions of these permutations in the "canonical list" A055089 (where also the rest of procedures can be found). A060118 gives position of the inverse permutation of each and A065183 positions after Foata transform.
Inversion vectors: A064039.
Cf. A060125, A060128, A060129, A060130, A060131, A060132, A060495.

with(group); permul := (a,b) -> mulperms(b,a); PermUnrank3R := proc(r) local n; n := nops(factorial_base(r)); convert(PermUnrank3Raux(n+1,r,[]),'permlist',1+(((r+2) mod (r+1))*n)); end; PermUnrank3Raux := proc(n,r,p) local s; if(0 = r) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Raux(n-1, r-(s*((n-1)!)), permul(p,[[n,n-s]]))); fi; end;

[seq(op(PermUnrank3R(j)), j=0..)]; (Maple code given below)

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A007623 Integers written in factorial base.

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A034968 Minimal number of factorials that add to n.

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A060117 A list of all finite permutations in "PermUnrank3R" ordering. (Inverses of the permutations of A060118.)

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