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A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n. 8
1, 1, 1, 1, 2, 3, 6, 4, 15, 11, 20, 21, 90, 51, 80, 32, 468, 166, 2910, 124, 521, 277, 20644, 266, 621, 1761, 1866, 841, 165874, 1374, 1484344, 436, 3797, 12741, 5383, 3108, 14653890, 103783, 31323, 2294, 158136988, 12419, 1852077284, 6382, 20786, 939131, 23394406084 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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.

LINKS

Table of n, a(n) for n=1..47.

FORMULA

a(n) = A318812(A181821(n)).

a(prime(n)) = A318813(n).

a(2^n) = A005121(n).

EXAMPLE

The a(12) = 21 multisystems on {1,1,2,3} (commas elided):

  {1123}  {{1}{123}}  {{1}{1}{23}}  {{{1}}{{1}{23}}}

          {{2}{113}}  {{1}{2}{13}}  {{{23}}{{1}{1}}}

          {{3}{112}}  {{1}{3}{12}}  {{{1}}{{2}{13}}}

          {{11}{23}}  {{2}{3}{11}}  {{{2}}{{1}{13}}}

          {{12}{13}}                {{{13}}{{1}{2}}}

                                    {{{1}}{{3}{12}}}

                                    {{{3}}{{1}{12}}}

                                    {{{12}}{{1}{3}}}

                                    {{{2}}{{3}{11}}}

                                    {{{3}}{{2}{11}}}

                                    {{{11}}{{2}{3}}}

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];

tmsp[m_]:=Prepend[Join@@Table[tmsp[c], {c, Select[mps[m], 1<Length[#]<Length[m]&]}], m];

Table[Length[tmsp[nrmptn[n]]], {n, 20}]

CROSSREFS

Row sums of A330727.

Cf. A001055, A002846, A005121, A181821, A213427, A281118, A281119, A305936, A318762, A318812, A318813.

Cf. A318847, A318848, A318849.

Sequence in context: A285331 A237125 A227296 * A231263 A231451 A126063

Adjacent sequences:  A318843 A318844 A318845 * A318847 A318848 A318849

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 04 2018

EXTENSIONS

Terminology corrected by Gus Wiseman, Jan 04 2020

More terms from Jinyuan Wang, Jun 26 2020

STATUS

approved

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Last modified July 27 15:12 EDT 2021. Contains 346307 sequences. (Running on oeis4.)