A356945 - OEIS

%I #8 Sep 10 2022 22:01:57

%S 1,1,0,2,0,1,0,3,0,0,0,2,0,0,0,5,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,7,0,0,

%T 0,3,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,11,0,0,0,0,

%U 0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.

%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.

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vR-C_picqWlu0KOguRGWaPjhS2HY7m43aGXGDcolDh4Qtyy-pu2lkq5mbHAbiMSyQoiIESG2mCGtc2j/pub">Counting and ranking classes of multiset partitions related to gapless multisets</a>

%e The a{n} multiset partitions for n = 8, 24, 72, 96:

%e   {{111}}      {{1112}}      {{11122}}      {{111112}}

%e   {{1}{11}}    {{1}{112}}    {{1}{1122}}    {{1}{11112}}

%e   {{1}{1}{1}}  {{11}{12}}    {{11}{122}}    {{11}{1112}}

%e                {{1}{1}{12}}  {{12}{112}}    {{111}{112}}

%e                              {{1}{1}{122}}  {{12}{1111}}

%e                              {{1}{12}{12}}  {{1}{1}{1112}}

%e                                             {{1}{11}{112}}

%e                                             {{11}{11}{12}}

%e                                             {{1}{12}{111}}

%e                                             {{1}{1}{1}{112}}

%e                                             {{1}{1}{11}{12}}

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

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

%t nnQ[m_]:=PrimePi/@First/@FactorInteger[m]==Range[PrimePi[Max@@First/@FactorInteger[m]]];

%t Table[Length[Select[facs[n],And@@nnQ/@#&]],{n,100}]

%Y Positions of 0's are A080259, complement A055932.

%Y A000688 counts factorizations into prime powers.

%Y A001055 counts factorizations.

%Y A001221 counts prime divisors, with sum A001414.

%Y A001222 counts prime factors with multiplicity.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A356069 counts gapless divisors, initial A356224 (complement A356225).

%Y Multisets covering an initial interval are counted by A000009, A000041, A011782, ranked by A055932.

%Y Other types: A034691, A089259, A356954, A356955.

%Y Other conditions: A050320, A050330, A322585, A356233, A356931, A356936.

%Y Cf. A003963, A073491, A107742, A287170, A324929, A340852, A356234.

%K nonn

%O 1,4

%A _Gus Wiseman_, Sep 08 2022