A050382 -id:A050382 - cp's OEIS frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

2, 3, 10, 4, 12, 6, 5, 15, 20, 420, 7, 21, 30, 630, 120, 8, 24, 56, 840, 1680, 210, 9, 28, 60, 1980, 60060, 1260, 4324320, 11, 35, 90, 3003, 83160, 2520, 21621600, 7207200, 13, 36, 105, 7140, 180180, 5040, 24504480, 151351200, 720720

Offset: 1

Pontus von Brömssen, Sep 22 2024

Row n lists numbers m such that A376369(m) = n.
In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.
Any integer k >= 2 appears exactly once in the array.

			Array begins:
  n\k|       1         2         3         4         5          6          7          8
  ---+---------------------------------------------------------------------------------
  1  |       2         3         4         5         7          8          9         11
  2  |      10        12        15        21        24         28         35         36
  3  |       6        20        30        56        60         90        105        252
  4  |     420       630       840      1980      3003       7140       7560       9240
  5  |     120      1680     60060     83160    180180     240240     831600     900900
  6  |     210      1260      2520      5040     27720     166320    1441440    4084080
  7  | 4324320  21621600  24504480  43243200  75675600  116396280  367567200  908107200
  8  | 7207200 151351200 302702400 411863760 823727520 1816214400 2327925600 4655851200

Pontus von Brömssen, Table of n, a(n) for n = 1..1081 (antidiagonals 1..46)

Cf. A036038, A050382, A078760, A318762, A325472 (complement of first row), A325593 (complement of the union of the first 2 rows), A376369, A376376 (first column).

First five rows are A376371, A376372, A376373, A376374, A376375.

A008480 Number of ordered prime factorizations of n.

Original entry on oeis.org

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

Olivier Gérard

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

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.

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.

Cf. A000040, A000142, A000961, A002110, A002033, A050382.
Cf. A036038, A036039, A036040, A080575, A102189.
Cf. A099848, A099849, A112624, A130675.
Cf. A124010, record values and where they occur: A260987, A260633.
Absolute values of A355939.

a008480 n = foldl div (a000142 $ sum es) (map a000142 es)
            where es = a124010_row n
-- Reinhard Zumkeller, Nov 18 2015

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

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 *)

a(n)={my(sig=factor(n)[,2]); vecsum(sig)!/vecprod(apply(k->k!, sig))} \\ Andrew Howroyd, Nov 17 2018

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

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

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
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
a(n) = abs(A355939(n)). - Antti Karttunen and Vaclav Kotesovec, Jul 22 2022
a(n) = A130675(n)/A112624(n). - Amiram Eldar, Mar 08 2024

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Pontus von Brömssen, Sep 22 2024

nonn

a(n) is the number of occurrences of n in each of A036038, A050382, A078760, A318762, and A376367.

The sequence is unbounded. To see this, note that the sets of parts (1,1,1,4) and (2,2,3) of a partition can be exchanged without affecting the value of the multinomial coefficient, because 1+1+1+4 = 2+2+3 and 1!*1!*1!*4! = 2!*2!*3!. In particular, a((7*k)!/24^k) >= k+1 from the partitions 7*k = (3*j)*1 + j*4 + (2*(k-j))*2 + (k-j)*3 for 0 <= j <= k.

			a(6) = 3, because 6 can be written as a multinomial coefficient in 3 ways: 6 = 6!/(1!*5!) = 4!/(2!*2!) = 3!/(1!*1!*1!).

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A036038, A050382, A078760, A318762, A376367, A376368, A376370.

cp's OEIS Frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

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A008480 Number of ordered prime factorizations of n.

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A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

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A376367 Sorted multinomial coefficients greater than 1, including duplicates.

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A329889 a(n) is the unique integer k such that A108951(k) = A260633(n).

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