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

Author

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

Keywords

• base
• nonn
• nice
• easy

Comments

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

Examples

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

References

• 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).

Links

• 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

Crossrefs

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.

Programs

Haskell

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Extensions

More terms from R. K. Guy