A330667 - OEIS

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 #6 Dec 31 2019 08:24:23

%S 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 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 0,1,0,1,7,7,0,1,1,0,1,0,1,5,5,0,1,1,3

%N 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 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 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 Triangle begins:

%e {}

%e 1

%e 1 0

%e 1

%e 1 0

%e 1

%e 1 1 0

%e 1 0

%e 1

%e 1 2 0

%e 1

%e 1 0

%e 1 3 2 0

%e 1

%e 1 2 0

%e 1

%e 1 2 0

%e Row n = 84 counts the following multisystems (commas elided):

%e {1124} {{1}{124}} {{{1}}{{1}{24}}}

%e {{11}{24}} {{{11}}{{2}{4}}}

%e {{12}{14}} {{{1}}{{2}{14}}}

%e {{2}{114}} {{{12}}{{1}{4}}}

%e {{4}{112}} {{{1}}{{4}{12}}}

%e {{1}{1}{24}} {{{14}}{{1}{2}}}

%e {{1}{2}{14}} {{{2}}{{1}{14}}}

%e {{1}{4}{12}} {{{2}}{{4}{11}}}

%e {{2}{4}{11}} {{{24}}{{1}{1}}}

%e {{{4}}{{1}{12}}}

%e {{{4}}{{2}{11}}}

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t totfac[n_,k_]:=If[k==1,1,Sum[totfac[Times@@Prime/@f,k-1],{f,Select[facs[n],1<Length[#]<PrimeOmega[n]&]}]];

%t Table[totfac[n,k],{n,100},{k,PrimeOmega[n]}]

%Y Row lengths are A001222.

%Y Row sums are A318812.

%Y The last nonzero term of row n is A330665(n).

%Y Column k = 2 is 0 if n is prime; otherwise it is A001055(n) - 2.

%Y Cf. A000311, A000669, A001678, A005121, A008827, A213427, A317145, A318846, A330474, A330475, A330655, A330666.

%K nonn,tabf

%O 1,18

%A _Gus Wiseman_, Dec 27 2019