A343935 - OEIS

%I #10 Jul 05 2024 09:31:50

%S 1,3,4,15,6,84,8,165,55,286,12,6188,14,680,816,4845,18,33649,20,53130,

%T 2024,2300,24,2629575,351,3654,4060,237336,30,10295472,32,435897,7140,

%U 7770,8436,177232627,38,10660,11480,62891499,42,85900584,44,1906884,2118760

%N Number of ways to choose a multiset of n divisors of n.

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

%e The a(1) = 1 through a(5) = 6 multisets:

%e   {1}  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}

%e        {1,2}  {1,1,3}  {1,1,1,2}  {1,1,1,1,5}

%e        {2,2}  {1,3,3}  {1,1,1,4}  {1,1,1,5,5}

%e               {3,3,3}  {1,1,2,2}  {1,1,5,5,5}

%e                        {1,1,2,4}  {1,5,5,5,5}

%e                        {1,1,4,4}  {5,5,5,5,5}

%e                        {1,2,2,2}

%e                        {1,2,2,4}

%e                        {1,2,4,4}

%e                        {1,4,4,4}

%e                        {2,2,2,2}

%e                        {2,2,2,4}

%e                        {2,2,4,4}

%e                        {2,4,4,4}

%e                        {4,4,4,4}

%t multchoo[n_,k_]:=Binomial[n+k-1,k];

%t Table[multchoo[DivisorSigma[0,n],n],{n,25}]

%o (Python)

%o from math import comb

%o from sympy import divisor_count

%o def A343935(n): return comb(divisor_count(n)+n-1,n) # _Chai Wah Wu_, Jul 05 2024

%Y Diagonal n = k of A343658.

%Y Choosing n divisors of n - 1 gives A343936.

%Y The version for chains of divisors is A343939.

%Y A000005 counts divisors.

%Y A000312 = n^n.

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

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

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

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

%Y A253249 counts nonempty chains of divisors of n.

%Y Strict chains of divisors:

%Y - A067824 counts strict chains of divisors starting with n.

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

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

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

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

%Y - A337256 counts strict chains of divisors of n.

%Y - A343662 counts strict length-k chains of divisors.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, May 05 2021