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A330667
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Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose atoms are the prime indices of n.
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4
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1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 1, 0, 1, 0, 1, 3, 2, 0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 3, 0, 1, 1, 5, 9, 5, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 0, 1, 1, 0, 1, 0, 1, 5, 5, 0, 1, 1, 3
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OFFSET
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1,18
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COMMENTS
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A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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LINKS
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EXAMPLE
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Triangle begins:
{}
1
1
1 0
1
1 0
1
1 1 0
1 0
1 0
1
1 2 0
1
1 0
1 0
1 3 2 0
1
1 2 0
1
1 2 0
Row n = 84 counts the following multisystems (commas elided):
{1124} {{1}{124}} {{{1}}{{1}{24}}}
{{11}{24}} {{{11}}{{2}{4}}}
{{12}{14}} {{{1}}{{2}{14}}}
{{2}{114}} {{{12}}{{1}{4}}}
{{4}{112}} {{{1}}{{4}{12}}}
{{1}{1}{24}} {{{14}}{{1}{2}}}
{{1}{2}{14}} {{{2}}{{1}{14}}}
{{1}{4}{12}} {{{2}}{{4}{11}}}
{{2}{4}{11}} {{{24}}{{1}{1}}}
{{{4}}{{1}{12}}}
{{{4}}{{2}{11}}}
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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_, k_]:=If[k==1, 1, Sum[totfac[Times@@Prime/@f, k-1], {f, Select[facs[n], 1<Length[#]<PrimeOmega[n]&]}]];
Table[totfac[n, k], {n, 100}, {k, PrimeOmega[n]}]
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CROSSREFS
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The last nonzero term of row n is A330665(n).
Column k = 2 is 0 if n is prime; otherwise it is A001055(n) - 2.
Cf. A000311, A000669, A001678, A005121, A008827, A213427, A317145, A318846, A330474, A330475, A330655, A330666.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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