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A007623
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Integers written in factorial base.
(Formerly M4678)
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220
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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.
Wikipedia, Factorial base
Index entries for sequences related to factorial base representation
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EXAMPLE
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a(47) = 1321 because 47 = 1*4! + 3*3! + 2*2! + 1*1!
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MAPLE
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a := n -> if nargs<2 then a(n, 2) elif n<args[2] then n else a(iquo(n, args[2]), args[2]+1)*10+irem(n, args[2]) fi: 'a(i)'$i=0..200;
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MATHEMATICA
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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 *)
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PROG
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(PARI) apply( a(n, p=2)=if(n<p, n, a(n\p, p+1)*10 + n%p)), [0..199]) \\ M. F. Hasler, Mar 27 2007; minor edit Nov 26 2018
(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
(Python)
def a(n, p=2): return n if n<p else a(int(n/p), p+1)*10 + n%p
print [a(n - 1) for n in xrange(1, 201)] # Indranil Ghosh, Jun 19 2017, after PARI program
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CROSSREFS
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Cf. A000142.
Cf. A034968 (sum of digits).
Cf. A060130 (number of nonzero digits).
Cf. 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.
Compare also to primorial base A049345.
Sequence in context: A261909 A235202 A049345 * A109827 A109839 A280149
Adjacent sequences: A007620 A007621 A007622 * A007624 A007625 A007626
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KEYWORD
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base,nonn,nice,easy
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein
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EXTENSIONS
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More terms from R. K. Guy
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STATUS
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approved
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