A343935 Number of ways to choose a multiset of n divisors of n.

1, 3, 4, 15, 6, 84, 8, 165, 55, 286, 12, 6188, 14, 680, 816, 4845, 18, 33649, 20, 53130, 2024, 2300, 24, 2629575, 351, 3654, 4060, 237336, 30, 10295472, 32, 435897, 7140, 7770, 8436, 177232627, 38, 10660, 11480, 62891499, 42, 85900584, 44, 1906884, 2118760

Offset

1,2

Links

Table of n, a(n) for n=1..45.

Formula

a(n) = ((sigma(n), n)) = binomial(sigma(n) + n - 1, n) where sigma = A000005 and binomial = A007318.

Example

The a(1) = 1 through a(5) = 6 multisets:
  {1}  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}
       {1,2}  {1,1,3}  {1,1,1,2}  {1,1,1,1,5}
       {2,2}  {1,3,3}  {1,1,1,4}  {1,1,1,5,5}
              {3,3,3}  {1,1,2,2}  {1,1,5,5,5}
                       {1,1,2,4}  {1,5,5,5,5}
                       {1,1,4,4}  {5,5,5,5,5}
                       {1,2,2,2}
                       {1,2,2,4}
                       {1,2,4,4}
                       {1,4,4,4}
                       {2,2,2,2}
                       {2,2,2,4}
                       {2,2,4,4}
                       {2,4,4,4}
                       {4,4,4,4}

Mathematica

multchoo[n_, k_]:=Binomial[n+k-1, k];
Table[multchoo[DivisorSigma[0, n], n], {n, 25}]

Prog

(Python)
from math import comb
from sympy import divisor_count
def A343935(n): return comb(divisor_count(n)+n-1, n) # Chai Wah Wu, Jul 05 2024

Crossrefs

Diagonal n = k of A343658.

Choosing n divisors of n - 1 gives A343936.

The version for chains of divisors is A343939.

A000005 counts divisors.

A000312 = n^n.

A007318 counts k-sets of elements of {1..n}.

A009998 = n^k (as an array, offset 1).

A059481 counts k-multisets of elements of {1..n}.

A146291 counts divisors of n with k prime factors (with multiplicity).

A253249 counts nonempty chains of divisors of n.

Strict chains of divisors:

- A067824 counts strict chains of divisors starting with n.

- A074206 counts strict chains of divisors from n to 1.

- A251683 counts strict length k + 1 chains of divisors from n to 1.

- A334996 counts strict length-k chains of divisors from n to 1.

- A337255 counts strict length-k chains of divisors starting with n.

- A337256 counts strict chains of divisors of n.

- A343662 counts strict length-k chains of divisors.

Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343661.

Sequence in context: A071167 A194924 A205489 * A280685 A076365 A070971

Adjacent sequences: A343932 A343933 A343934 * A343936 A343937 A343938

Keyword

nonn

Author

Gus Wiseman, May 05 2021

Status

approved