A008480 Number of ordered prime factorizations of n.

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

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

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

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. 151-165.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.

Links

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

Formula

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

a(n) = abs(A355939(n)). - Antti Karttunen and Vaclav Kotesovec, Jul 22 2022

Maple

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

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] (* Jean-François Alcover, Sep 15 2019 *)

Prog

(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
(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

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.

Absolute values of A355939.

Sequence in context: A335521 A323087 A321747 * A168324 A355939 A303838

Adjacent sequences: A008477 A008478 A008479 * A008481 A008482 A008483

Keyword

nonn

Author

Olivier Gérard

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

Status

approved