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
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.
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
Links
- T. D. Noe, Table of n, a(n) for n = 1..10000
- Steven 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, Exact formulas for partial sums of the Möbius function expressed by partial sums weighted by the Liouville lambda function, arXiv:2102.05842 [math.NT], 2021-2022.
- Eric Weisstein's World of Mathematics, Multinomial Coefficient.
Crossrefs
Programs
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Haskell
a008480 n = foldl div (a000142 $ sum es) (map a000142 es) where es = a124010_row n -- Reinhard Zumkeller, Nov 18 2015 -
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
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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 *) -
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
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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
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(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
Extensions
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007
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