cp's OEIS Frontend

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.

%I A330667 #6 Dec 31 2019 08:24:23
%S A330667 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,
%T A330667 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,
%U A330667 0,1,0,1,7,7,0,1,1,0,1,0,1,5,5,0,1,1,3
%N 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.
%C A330667 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.
%C A330667 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.
%e A330667 Triangle begins:
%e A330667   {}
%e A330667   1
%e A330667   1
%e A330667   1 0
%e A330667   1
%e A330667   1 0
%e A330667   1
%e A330667   1 1 0
%e A330667   1 0
%e A330667   1 0
%e A330667   1
%e A330667   1 2 0
%e A330667   1
%e A330667   1 0
%e A330667   1 0
%e A330667   1 3 2 0
%e A330667   1
%e A330667   1 2 0
%e A330667   1
%e A330667   1 2 0
%e A330667 Row n = 84 counts the following multisystems (commas elided):
%e A330667   {1124}  {{1}{124}}    {{{1}}{{1}{24}}}
%e A330667           {{11}{24}}    {{{11}}{{2}{4}}}
%e A330667           {{12}{14}}    {{{1}}{{2}{14}}}
%e A330667           {{2}{114}}    {{{12}}{{1}{4}}}
%e A330667           {{4}{112}}    {{{1}}{{4}{12}}}
%e A330667           {{1}{1}{24}}  {{{14}}{{1}{2}}}
%e A330667           {{1}{2}{14}}  {{{2}}{{1}{14}}}
%e A330667           {{1}{4}{12}}  {{{2}}{{4}{11}}}
%e A330667           {{2}{4}{11}}  {{{24}}{{1}{1}}}
%e A330667                         {{{4}}{{1}{12}}}
%e A330667                         {{{4}}{{2}{11}}}
%t A330667 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A330667 totfac[n_,k_]:=If[k==1,1,Sum[totfac[Times@@Prime/@f,k-1],{f,Select[facs[n],1<Length[#]<PrimeOmega[n]&]}]];
%t A330667 Table[totfac[n,k],{n,100},{k,PrimeOmega[n]}]
%Y A330667 Row lengths are A001222.
%Y A330667 Row sums are A318812.
%Y A330667 The last nonzero term of row n is A330665(n).
%Y A330667 Column k = 2 is 0 if n is prime; otherwise it is A001055(n) - 2.
%Y A330667 Cf. A000311, A000669, A001678, A005121, A008827, A213427, A317145, A318846, A330474, A330475, A330655, A330666.
%K A330667 nonn,tabf
%O A330667 1,18
%A A330667 _Gus Wiseman_, Dec 27 2019