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 A008480 Number of ordered prime factorizations of n. 134
 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 (list; graph; refs; listen; history; text; internal format)
 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] Gordon Hamilton's MathPickle, Fractal Multiplication (visual presentation of non-commutative multiplication) 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 MAPLE a:= n-> (l-> add(i, i=l)!/mul(i!, i=l))(map(i-> i, ifactors(n))): 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 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 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. Sequence in context: A303707 A323087 A321747 * A168324 A303838 A324837 Adjacent sequences:  A008477 A008478 A008479 * A008481 A008482 A008483 KEYWORD nonn AUTHOR EXTENSIONS Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007 STATUS approved

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Last modified May 24 19:14 EDT 2020. Contains 334580 sequences. (Running on oeis4.)