

A008480


Number of ordered prime factorizations of n.


203



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

1,6


COMMENTS

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
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


REFERENCES

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. 151165.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292295.


LINKS



FORMULA

If n = Product (p_j^k_j) then a(n) = ( Sum (k_j) )! / Product (k_j !).
Dirichlet g.f.: 1/(1B(s)) where B(s) is D.g.f. of characteristic function of primes.
a(p^k) = 1 if p is a prime.
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 kfold 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) * (k1)!).  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 CLTtype ErdősKac theorem analog proved by M. D. Schmidt, 2021.  Maxie D. Schmidt, Feb 12 2021


MAPLE

a:= n> (l> add(i, i=l)!/mul(i!, i=l))(map(i> i[2], ifactors(n)[2])):


MATHEMATICA

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] (* JeanFrançois Alcover, Sep 15 2019 *)


PROG

(Sage)
S = [s[1] for s in factor(n)]
return factorial(sum(S)) // prod(factorial(s) for s in S)
(Haskell)
a008480 n = foldl div (a000142 $ sum es) (map a000142 es)
where es = a124010_row n
(PARI) a(n)={my(sig=factor(n)[, 2]); vecsum(sig)!/vecprod(apply(k>k!, sig))} \\ Andrew Howroyd, Nov 17 2018
(Python)
from math import prod, factorial
from sympy import factorint
def A008480(n): return factorial(sum(f:=factorint(n).values()))//prod(map(factorial, f)) # Chai Wah Wu, Aug 05 2023


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



