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A008480
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Number of ordered prime factorizations of n.
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198
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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
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OFFSET
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1,6
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COMMENTS
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a(n) depends only on the 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, Nov 07 2006
The Dirichlet inverse is given by A080339, negating all but the A080339(1) element in A080339. - R. J. Mathar, Jul 15 2010
Number of (distinct) permutations of the multiset of prime factors. - Joerg Arndt, Feb 17 2015
Number of not divisible chains in the divisor lattice of n. - Peter Luschny, Jun 15 2013
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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, June 5, 2003. [Cached copy, with permission of the author]
Carl-Erik Fröberg, On the prime zeta function, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202.
Gordon Hamilton's MathPickle, Fractal Multiplication (visual presentation of non-commutative multiplication).
Maxie D. Schmidt, New characterizations of the summatory function of the Moebius function, arXiv:2102.05842 [math.NT], 2021.
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.
a(p^k) = 1 if p is a prime.
a(A002110(n)) = A000142(n) = n!.
a(n) = A050382(A101296(n)). - R. J. Mathar, May 26 2017
a(n) = 1 <=> n in { A000961 }. - Alois P. Heinz, May 26 2018
G.f. A(x) satisfies: A(x) = x + A(x^2) + A(x^3) + A(x^5) + ... + A(x^prime(k)) + ... - Ilya Gutkovskiy, May 10 2019
a(n) = C(k, n) for k = A001222(n) where C(k, n) is defined as the k-fold Dirichlet convolution of A001221(n) with itself, and where C(0, n) is the multiplicative identity with respect to Dirichlet convolution.
The average order of a(n) is asymptotic (up to an absolute constant) to 2A sqrt(2*Pi) log(n) / sqrt(log(log(n))) for some absolute constant A > 0. - Maxie D. Schmidt, May 28 2021
The sums of a(n) for n <= x and k >= 1 such that A001222(n)=k have asymptotic order of the form x*(log(log(x)))^(k+1/2) / ((2k+1) * (k-1)!). - Maxie D. Schmidt, Feb 12 2021
Other DGFs include: (1+P(s))^(-1) in terms of the prime zeta function for Re(s) > 1 where the + version weights the sequence by A008836(n), see the reference by Fröberg on P(s). - Maxie D. Schmidt, Feb 12 2021
The bivariate DGF (1+zP(s))^(-1) has coefficients a(n) / n^s (-1)^(A001221(n)) z^(A001222(n)) for Re(s) > 1 and 0 < |z| < 2 - Maxie D. Schmidt, Feb 12 2021
The distribution of the distinct values of the sequence for n<=x as x->infinity satisfy a CLT-type Erdős-Kac theorem analog proved by M. D. Schmidt, 2021. - Maxie D. Schmidt, Feb 12 2021
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MAPLE
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a:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):
seq(a(n), n=1..100); # Alois P. Heinz, May 26 2018
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MATHEMATICA
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Prepend[ Array[ Multinomial @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
(* Second program: *)
a[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times @@ (ee!)]; Array[a, 101] (* Jean-François Alcover, Sep 15 2019 *)
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PROG
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(Sage)
def A008480(n):
S = [s[1] for s in factor(n)]
return factorial(sum(S)) // prod(factorial(s) for s in S)
[A008480(n) for n in (1..101)] # Peter Luschny, Jun 15 2013
(Haskell)
a008480 n = foldl div (a000142 $ sum es) (map a000142 es)
where es = a124010_row n
-- Reinhard Zumkeller, Nov 18 2015
(PARI) a(n)={my(sig=factor(n)[, 2]); vecsum(sig)!/vecprod(apply(k->k!, sig))} \\ Andrew Howroyd, Nov 17 2018
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CROSSREFS
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Cf. A000040, A000142, A000961, A002110, A002033, A050382.
Cf. A036038, A036039, A036040, A080575, A102189.
Cf. A099848, A099849.
Cf. A124010, record values and where they occur: A260987, A260633.
Sequence in context: A335521 A323087 A321747 * A168324 A303838 A324837
Adjacent sequences: A008477 A008478 A008479 * A008481 A008482 A008483
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KEYWORD
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nonn
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AUTHOR
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Olivier Gérard
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EXTENSIONS
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Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007
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STATUS
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approved
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