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%I A008480 #123 Feb 16 2025 08:32:32
%S A008480 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 A008480 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 A008480 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 A008480 Number of ordered prime factorizations of n.
%C A008480 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 A008480 Multinomial coefficients in prime factorization order. - _Max Alekseyev_, Nov 07 2006
%C A008480 The Dirichlet inverse is given by A080339, negating all but the A080339(1) element in A080339. - _R. J. Mathar_, Jul 15 2010
%C A008480 Number of (distinct) permutations of the multiset of prime factors. - _Joerg Arndt_, Feb 17 2015
%C A008480 Number of not divisible chains in the divisor lattice of n. - _Peter Luschny_, Jun 15 2013
%D A008480 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 A008480 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
%H A008480 T. D. Noe, <a href="/A008480/b008480.txt">Table of n, a(n) for n = 1..10000</a>
%H A008480 Steven R. Finch, <a href="/A001055/a001055.pdf">Kalmar's composition constant</a>, June 5, 2003. [Cached copy, with permission of the author]
%H A008480 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 A008480 Gordon Hamilton's MathPickle, <a href="http://mathpickle.com/project/fractal-multiplication/">Fractal Multiplication</a> (visual presentation of non-commutative multiplication).
%H A008480 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 A008480 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultinomialCoefficient.html">Multinomial Coefficient</a>.
%F A008480 If n = Product (p_j^k_j) then a(n) = ( Sum (k_j) )! / Product (k_j !).
%F A008480 Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of primes.
%F A008480 a(p^k) = 1 if p is a prime.
%F A008480 a(A002110(n)) = A000142(n) = n!.
%F A008480 a(n) = A050382(A101296(n)). - _R. J. Mathar_, May 26 2017
%F A008480 a(n) = 1 <=> n in { A000961 }. - _Alois P. Heinz_, May 26 2018
%F A008480 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 A008480 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 A008480 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 A008480 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 A008480 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 A008480 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 A008480 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 A008480 a(n) = abs(A355939(n)). - _Antti Karttunen_ and _Vaclav Kotesovec_, Jul 22 2022
%F A008480 a(n) = A130675(n)/A112624(n). - _Amiram Eldar_, Mar 08 2024
%p A008480 a:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i[2], ifactors(n)[2])):
%p A008480 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 26 2018
%t A008480 Prepend[ Array[ Multinomial @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
%t A008480 (* Second program: *)
%t A008480 a[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Total[ee]!/Times @@ (ee!)]; Array[a, 101] (* _Jean-François Alcover_, Sep 15 2019 *)
%o A008480 (Sage)
%o A008480 def A008480(n):
%o A008480     S = [s[1] for s in factor(n)]
%o A008480     return factorial(sum(S)) // prod(factorial(s) for s in S)
%o A008480 [A008480(n) for n in (1..101)]  # _Peter Luschny_, Jun 15 2013
%o A008480 (Haskell)
%o A008480 a008480 n = foldl div (a000142 $ sum es) (map a000142 es)
%o A008480             where es = a124010_row n
%o A008480 -- _Reinhard Zumkeller_, Nov 18 2015
%o A008480 (PARI) a(n)={my(sig=factor(n)[,2]); vecsum(sig)!/vecprod(apply(k->k!, sig))} \\ _Andrew Howroyd_, Nov 17 2018
%o A008480 (Python)
%o A008480 from math import prod, factorial
%o A008480 from sympy import factorint
%o A008480 def A008480(n): return factorial(sum(f:=factorint(n).values()))//prod(map(factorial,f)) # _Chai Wah Wu_, Aug 05 2023
%Y A008480 Cf. A000040, A000142, A000961, A002110, A002033, A050382.
%Y A008480 Cf. A036038, A036039, A036040, A080575, A102189.
%Y A008480 Cf. A099848, A099849, A112624, A130675.
%Y A008480 Cf. A124010, record values and where they occur: A260987, A260633.
%Y A008480 Absolute values of A355939.
%K A008480 nonn,easy,changed
%O A008480 1,6
%A A008480 _Olivier Gérard_
%E A008480 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 17 2007

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