A343936 Number of ways to choose a multiset of n divisors of n - 1.

1, 2, 3, 10, 5, 56, 7, 120, 45, 220, 11, 4368, 13, 560, 680, 3876, 17, 26334, 19, 42504, 1771, 2024, 23, 2035800, 325, 3276, 3654, 201376, 29, 8347680, 31, 376992, 6545, 7140, 7770, 145008513, 37, 9880, 10660, 53524680, 41, 73629072, 43, 1712304, 1906884

Offset

1,2

Links

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

Formula

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

Example

The a(1) = 1 through a(5) = 5 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
      {2}  {1,3}  {1,1,2}  {1,1,1,5}
           {3,3}  {1,1,4}  {1,1,5,5}
                  {1,2,2}  {1,5,5,5}
                  {1,2,4}  {5,5,5,5}
                  {1,4,4}
                  {2,2,2}
                  {2,2,4}
                  {2,4,4}
                  {4,4,4}
The a(6) = 56 multisets:
  11111  11136  11333  12236  13366  22266  23666
  11112  11166  11336  12266  13666  22333  26666
  11113  11222  11366  12333  16666  22336  33333
  11116  11223  11666  12336  22222  22366  33336
  11122  11226  12222  12366  22223  22666  33366
  11123  11233  12223  12666  22226  23333  33666
  11126  11236  12226  13333  22233  23336  36666
  11133  11266  12233  13336  22236  23366  66666

Mathematica

multchoo[n_, k_]:=Binomial[n+k-1, k];
Table[multchoo[DivisorSigma[0, n], n-1], {n, 50}]

Crossrefs

The version for chains of divisors is A163767.

Diagonal n = k + 1 of A343658.

Choosing n divisors of n gives A343935.

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, A343656, A343661.

Sequence in context: A334628 A292239 A163767 * A336091 A347835 A128531

Adjacent sequences: A343933 A343934 A343935 * A343937 A343938 A343939

Keyword

nonn

Author

Gus Wiseman, May 05 2021

Status

approved