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A007623
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Integers written in factorial base.
(Formerly M4678)
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102
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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;
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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 about 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
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REFERENCES
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Italo J. Dejter, A numeral system for the middle levels, Preprint 2014; http://home.coqui.net/dejterij/anumeral.pdf
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, Table of n, a(n) for n = 0..1000
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
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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 *)
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PROG
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(PARI) a(n, p=2) = if( n<p, n, a( n\p, p+1 )*10 + n%p ); vector(200, i, a(i-1)) \\ M. F. Hasler, Mar 27 2007
(Haskell)
a007623 n | n <= 36287999 = read $ concatMap show (a108731_row n) :: Int
| otherwise = error "representation would be ambiguous"
-- Reinhard Zumkeller, Jun 04 2012
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CROSSREFS
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Cf. A000142. See also primorial base A049345, sum of digits A034968, number of nonzero digits A060130. Simple algorithm fac_base given in A055089.
Cf. A060112, A060495. Permutation of A064039.
Sequence in context: A197652 A235202 A049345 * A109827 A109839 A087486
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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