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 A330667 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,18 COMMENTS 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. LINKS EXAMPLE 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}}} MATHEMATICA 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

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Last modified July 25 23:39 EDT 2021. Contains 346294 sequences. (Running on oeis4.)