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Search: id:A008480
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| A008480 |
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Number of ordered prime factorizations of n. |
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+0
8
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| 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 12, 1, 2, 3, 1, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 3, 3, 2, 6, 1, 5, 1, 2, 1, 12, 2, 2, 2, 4, 1, 12, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1).
Multinomial coefficients in prime factorization order. - Max Alekseyev (maxale(AT)gmail.com), Nov 07 2006
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REFERENCES
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A. Knopfmacher, J. Knopfmacher and R. Warlimont, "Ordered factorizations for integers and arithmetical semigroups", in _Advances in Number Theory_, (Proc. 3rd Conf. Canadian Number Theory Assoc., 1991), Clarendon Press, Oxford, 1993, pp. 151-165.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
S. R. Finch, Kalmar's composition constant
Eric Weisstein's World of Mathematics, Multinomial Coefficient
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FORMULA
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If n = Product (p_j^k_j) then a(n) = ( Sum (k_j) )! / Product (k_j !).
Dirichlet g.f.: 1/(1-B(s)) where B(s) is d.g.f. of characteristic function of primes.
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MATHEMATICA
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Prepend[ Array[ Multinomial @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
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CROSSREFS
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Cf. A000040, A002033, A050382. a(p^k)=1. a(A002110)=A000142=n!.
Cf. A036038, A036039, A036040, A080575, A102189.
Cf. A099848, A099849.
Sequence in context: A123529 A140747 A081707 this_sequence A066882 A068347 A025865
Adjacent sequences: A008477 A008478 A008479 this_sequence A008481 A008482 A008483
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KEYWORD
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nonn
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AUTHOR
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Olivier Gerard (olivier.gerard(AT)gmail.com)
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Plewe, Jun 17 2007
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