login
This site is supported by donations to The OEIS Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008480 Number of ordered prime factorizations of n. 109
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 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

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 ]

PROG

(Sage)

def A008480(n):

    S = [s[1] for s in list(factor(n))]

    return factorial(add(S)) / mul([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 A285572

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 19 18:16 EST 2019. Contains 319309 sequences. (Running on oeis4.)