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Search: id:A034968
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| A034968 |
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Minimal number of factorials which add to n. |
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+0
6
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| 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
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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Equivalently, sum of digits when n is written in factorial base (A007623).
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).
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FORMULA
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a(n)=n-sum(i>=2, (i-1)*floor(n/i!)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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!)) [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 13 2009]
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EXAMPLE
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a(205=1!*1+3!*2+4!*3+5!*1)=1+2+3+1=6.
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MAPLE
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[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)];
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PROGRAM
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(PARI) a(n)=local(k, r); k=2; r=0; while(n>0, r+=n%k; n\=k; k++); r [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 13 2009]
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CROSSREFS
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Partial sums of first n! terms: A001809. See A055091 for the minimum number of any transpositions. A034968[A056019[n]] = A034968[n] for all n.
Sequence in context: A097028 A092331 A089293 this_sequence A054707 A166269 A055460
Adjacent sequences: A034965 A034966 A034967 this_sequence A034969 A034970 A034971
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KEYWORD
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nonn
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AUTHOR
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Erich Friedman (erich.friedman(AT)stetson.edu)
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EXTENSIONS
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Additional comments from Antti Karttunen, Aug 23, 2001.
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