cp's OEIS Frontend

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

%I A318812 #11 Dec 31 2019 06:49:32
%S A318812 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,
%T A318812 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,
%U A318812 1,4,1,3,1,4,1,80,1,1,3,3,1,4,1,51,6,1,1
%N A318812 Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.
%C A318812 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.
%C A318812 a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Dec 30 2019
%H A318812 Andrew Howroyd, <a href="/A318812/b318812.txt">Table of n, a(n) for n = 1..10000</a>
%F A318812 a(product of n distinct primes) = A005121(n).
%F A318812 a(prime^n) = A318813(n).
%e A318812 The a(24) = 11 total multiset partitions:
%e A318812   {1,1,1,2}
%e A318812   {{1},{1,1,2}}
%e A318812   {{2},{1,1,1}}
%e A318812   {{1,1},{1,2}}
%e A318812   {{1},{1},{1,2}}
%e A318812   {{1},{2},{1,1}}
%e A318812   {{{1}},{{1},{1,2}}}
%e A318812   {{{1}},{{2},{1,1}}}
%e A318812   {{{2}},{{1},{1,1}}}
%e A318812   {{{1,2}},{{1},{1}}}
%e A318812   {{{1,1}},{{1},{2}}}
%e A318812 The a(24) = 11 total factorizations:
%e A318812   24,
%e A318812   (2*12), (3*8), (4*6),
%e A318812   (2*2*6), (2*3*4),
%e A318812   ((2)*(2*6)), ((6)*(2*2)), ((2)*(3*4)), ((3)*(2*4)), ((4)*(2*3)).
%t A318812 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A318812 totfac[n_]:=1+Sum[totfac[Times@@Prime/@f],{f,Select[facs[n],1<Length[#]<PrimeOmega[n]&]}];
%t A318812 Array[totfac,100]
%o A318812 (PARI)
%o A318812 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}
%o A318812 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
%Y A318812 Cf. A000110, A001055, A002846, A005121, A213427, A281113, A281118, A281119, A317145, A318813.
%K A318812 nonn
%O A318812 1,8
%A A318812 _Gus Wiseman_, Sep 04 2018