%I #122 Mar 08 2024 08:05:57
%S 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,
%T 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,
%U 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
%N Number of ordered prime factorizations of n.
%C 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).
%C Multinomial coefficients in prime factorization order. - _Max Alekseyev_, Nov 07 2006
%C The Dirichlet inverse is given by A080339, negating all but the A080339(1) element in A080339. - _R. J. Mathar_, Jul 15 2010
%C Number of (distinct) permutations of the multiset of prime factors. - _Joerg Arndt_, Feb 17 2015
%C Number of not divisible chains in the divisor lattice of n. - _Peter Luschny_, Jun 15 2013
%D 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.
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
%H T. D. Noe, <a href="/A008480/b008480.txt">Table of n, a(n) for n = 1..10000</a>
%H Steven R. Finch, <a href="/A001055/a001055.pdf">Kalmar's composition constant</a>, June 5, 2003. [Cached copy, with permission of the author]
%H Carl-Erik Fröberg, <a href="https://doi.org/10.1007/BF01933420">On the prime zeta function</a>, BIT Numerical Mathematics, Vol. 8, No. 3 (1968), pp. 187-202.
%H Gordon Hamilton's MathPickle, <a href="http://mathpickle.com/project/fractal-multiplication/">Fractal Multiplication</a> (visual presentation of non-commutative multiplication).
%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/2102.05842">Exact formulas for partial sums of the Möbius function expressed by partial sums weighted by the Liouville lambda function</a>, arXiv:2102.05842 [math.NT], 2021-2022.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultinomialCoefficient.html">Multinomial Coefficient</a>.
%F If n = Product (p_j^k_j) then a(n) = ( Sum (k_j) )! / Product (k_j !).
%F Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of primes.
%F a(p^k) = 1 if p is a prime.
%F a(A002110(n)) = A000142(n) = n!.
%F a(n) = A050382(A101296(n)). - _R. J. Mathar_, May 26 2017
%F a(n) = 1 <=> n in { A000961 }. - _Alois P. Heinz_, May 26 2018
%F 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
%F 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.
%F 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
%F 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
%F 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
%F 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
%F 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
%F a(n) = abs(A355939(n)). - _Antti Karttunen_ and _Vaclav Kotesovec_, Jul 22 2022
%F a(n) = A130675(n)/A112624(n). - _Amiram Eldar_, Mar 08 2024
%p a:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, May 26 2018
%t Prepend[ Array[ Multinomial @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
%t (* Second program: *)
%t a[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times @@ (ee!)]; Array[a, 101] (* _Jean-François Alcover_, Sep 15 2019 *)
%o (Sage)
%o def A008480(n):
%o S = [s[1] for s in factor(n)]
%o return factorial(sum(S)) // prod(factorial(s) for s in S)
%o [A008480(n) for n in (1..101)] # _Peter Luschny_, Jun 15 2013
%o (Haskell)
%o a008480 n = foldl div (a000142 $ sum es) (map a000142 es)
%o where es = a124010_row n
%o -- _Reinhard Zumkeller_, Nov 18 2015
%o (PARI) a(n)={my(sig=factor(n)[,2]); vecsum(sig)!/vecprod(apply(k->k!, sig))} \\ _Andrew Howroyd_, Nov 17 2018
%o (Python)
%o from math import prod, factorial
%o from sympy import factorint
%o def A008480(n): return factorial(sum(f:=factorint(n).values()))//prod(map(factorial,f)) # _Chai Wah Wu_, Aug 05 2023
%Y Cf. A000040, A000142, A000961, A002110, A002033, A050382.
%Y Cf. A036038, A036039, A036040, A080575, A102189.
%Y Cf. A099848, A099849, A112624, A130675.
%Y Cf. A124010, record values and where they occur: A260987, A260633.
%Y Absolute values of A355939.
%K nonn,easy
%O 1,6
%A _Olivier Gérard_
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 17 2007