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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

Table of n, a(n) for n=1..87.

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<Length[#]<PrimeOmega[n]&]}]];

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

CROSSREFS

Row lengths are A001222.

Row sums are A318812.

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.

Sequence in context: A291375 A033778 A091586 * A116377 A131964 A091430

Adjacent sequences:  A330664 A330665 A330666 * A330668 A330669 A330670

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Dec 27 2019

STATUS

approved

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