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 A318812 Number of total multiset partitions of the multiset of prime indices of n. Number of total factorizations of n. 31
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 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 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..10000 FORMULA a(product of n distinct primes) = A005121(n). a(prime^n) = A318813(n). EXAMPLE 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)). 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

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Last modified July 26 01:49 EDT 2021. Contains 346294 sequences. (Running on oeis4.)