A318812 Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 6, 1, 3, 1, 3, 1, 1, 1, 11, 1, 1, 2, 3, 1, 4, 1, 20, 1, 1, 1, 15, 1, 1, 1, 11, 1, 4, 1, 3, 3, 1, 1, 51, 1, 3, 1, 3, 1, 11, 1, 11, 1, 1, 1, 21, 1, 1, 3, 90, 1, 4, 1, 3, 1, 4, 1, 80, 1, 1, 3, 3, 1, 4, 1, 51, 6, 1, 1

Offset: 1

Author

Gus Wiseman, Sep 04 2018

Keywords

• nonn

Comments

A total multiset partition of m is either m itself or a total multiset partition of a multiset partition of m that is neither minimal nor maximal.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Dec 30 2019

Examples

			The a(24) = 11 total multiset partitions:
  {1,1,1,2}
  {{1},{1,1,2}}
  {{2},{1,1,1}}
  {{1,1},{1,2}}
  {{1},{1},{1,2}}
  {{1},{2},{1,1}}
  {{{1}},{{1},{1,2}}}
  {{{1}},{{2},{1,1}}}
  {{{2}},{{1},{1,1}}}
  {{{1,2}},{{1},{1}}}
  {{{1,1}},{{1},{2}}}
The a(24) = 11 total factorizations:
  24,
  (2*12), (3*8), (4*6),
  (2*2*6), (2*3*4),
  ((2)*(2*6)), ((6)*(2*2)), ((2)*(3*4)), ((3)*(2*4)), ((4)*(2*3)).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000110, A001055, A002846, A005121, A213427, A281113, A281118, A281119, A317145, A318813.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
totfac[n_]:=1+Sum[totfac[Times@@Prime/@f],{f,Select[facs[n],1
				

PARI

MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}
seq(n)={my(v=vector(n, i, isprime(i)), u=vector(n), m=logint(n,2)+1); for(r=1, m, u += v*sum(j=r, m, (-1)^(j-r)*binomial(j-1, r-1)); v=MultEulerT(v)); u[1]=1; u} \\ Andrew Howroyd, Dec 30 2019

Formula

a(product of n distinct primes) = A005121(n).

a(prime^n) = A318813(n).