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A318812
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Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n.
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31
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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
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OFFSET
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1,8
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COMMENTS
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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
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LINKS
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FORMULA
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a(product of n distinct primes) = A005121(n).
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EXAMPLE
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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)).
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MATHEMATICA
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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<Length[#]<PrimeOmega[n]&]}];
Array[totfac, 100]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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