A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.

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

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