cp's OEIS Frontend

A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.

%I A356931 #6 Sep 13 2022 13:06:56
%S A356931 1,1,0,2,1,0,0,3,0,2,1,0,0,0,0,5,1,0,0,4,0,2,1,0,2,0,0,0,0,0,1,7,0,2,
%T A356931 0,0,0,0,0,7,1,0,0,4,0,2,1,0,0,4,0,0,0,0,2,0,0,0,1,0,0,2,0,11,0,0,1,4,
%U A356931 0,0,0,0,1,0,0,0,0,0,0,12,0,2,1,0,2,0
%N A356931 Number of multiset partitions of the prime indices of n into multisets of odd numbers. Number of factorizations of n into members of A066208.
%C A356931 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 A356931 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>
%F A356931 a(n) = 0 if n is in A324929, otherwise a(n) = A001055(n).
%e A356931 The a(440) = 21 multiset partitions of {1,1,1,3,5}:
%e A356931   {1}{1}{1}{3}{5}  {1}{1}{1}{35}  {1}{1}{135}  {1}{1135}  {11135}
%e A356931                    {1}{1}{13}{5}  {1}{11}{35}  {11}{135}
%e A356931                    {1}{11}{3}{5}  {11}{13}{5}  {111}{35}
%e A356931                    {1}{1}{3}{15}  {1}{13}{15}  {113}{15}
%e A356931                                   {11}{3}{15}  {13}{115}
%e A356931                                   {1}{3}{115}  {3}{1115}
%e A356931                                   {1}{5}{113}  {5}{1113}
%e A356931                                   {3}{111}{5}
%t A356931 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356931 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A356931 Table[Length[Select[facs[n],And@@(OddQ[Times@@primeMS[#]]&/@#)&]],{n,100}]
%Y A356931 Positions of 0's are A324929, complement A066208.
%Y A356931 A000688 counts factorizations into prime powers.
%Y A356931 A001055 counts factorizations.
%Y A356931 A001221 counts prime divisors, sum A001414.
%Y A356931 A001222 counts prime factors with multiplicity.
%Y A356931 A056239 adds up prime indices, row sums of A112798.
%Y A356931 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356931 Other types: A000009, A000012, A000041, A089259, A356930.
%Y A356931 Other conditions: A050320, A050330, A356936, A322585, A356233, A356945.
%Y A356931 Cf. A003963, A073491, A107742, A287170, A325534, A325535, A340852, A356234.
%K A356931 nonn
%O A356931 1,4
%A A356931 _Gus Wiseman_, Sep 08 2022