cp's OEIS Frontend

A356936 Number of multiset partitions of the multiset of prime indices of n into intervals. Number of factorizations of n into members of A073485.

%I A356936 #8 Sep 10 2022 07:35:28
%S A356936 1,1,1,1,1,2,1,1,1,1,1,2,1,1,2,1,1,2,1,1,1,1,1,2,1,1,1,1,1,4,1,1,1,1,
%T A356936 2,3,1,1,1,1,1,2,1,1,2,1,1,2,1,1,1,1,1,2,1,1,1,1,1,4,1,1,1,1,1,2,1,1,
%U A356936 1,2,1,3,1,1,2,1,2,2,1,1,1,1,1,2,1,1,1
%N A356936 Number of multiset partitions of the multiset of prime indices of n into intervals. Number of factorizations of n into members of A073485.
%C A356936 An interval is a set of positive integers with all differences of adjacent elements equal to 1.
%C A356936 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 A356936 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 A356936 The a(n) multiset partitions for n = 6, 30, 36, 90, 180:
%e A356936   {12}    {123}      {12}{12}      {12}{23}      {12}{123}
%e A356936   {1}{2}  {1}{23}    {1}{2}{12}    {2}{123}      {1}{12}{23}
%e A356936           {3}{12}    {1}{1}{2}{2}  {1}{2}{23}    {1}{2}{123}
%e A356936           {1}{2}{3}                {2}{3}{12}    {3}{12}{12}
%e A356936                                    {1}{2}{2}{3}  {1}{1}{2}{23}
%e A356936                                                  {1}{2}{3}{12}
%e A356936                                                  {1}{1}{2}{2}{3}
%e A356936 The a(n) factorizations for n = 6, 30, 36, 90, 180:
%e A356936   (6)    (30)     (6*6)      (3*30)     (6*30)
%e A356936   (2*3)  (5*6)    (2*3*6)    (6*15)     (5*6*6)
%e A356936          (2*15)   (2*2*3*3)  (3*5*6)    (2*3*30)
%e A356936          (2*3*5)             (2*3*15)   (2*6*15)
%e A356936                              (2*3*3*5)  (2*3*5*6)
%e A356936                                         (2*2*3*15)
%e A356936                                         (2*2*3*3*5)
%t A356936 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356936 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A356936 chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
%t A356936 Table[Length[Select[facs[n],And@@chQ/@primeMS/@#&]],{n,100}]
%Y A356936 A000688 counts factorizations into prime powers.
%Y A356936 A001055 counts factorizations.
%Y A356936 A001221 counts prime divisors, sum A001414.
%Y A356936 A001222 counts prime factors with multiplicity.
%Y A356936 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356936 A056239 adds up prime indices, row sums of A112798.
%Y A356936 Intervals are counted by A000012, A001227, ranked by A073485.
%Y A356936 Other types: A107742, A356233, A356937, A356938, A356939.
%Y A356936 Other conditions: A050320, A050330, A322585, A356931, A356945.
%Y A356936 Cf. A003963, A073491, A287170, A328195, A356234.
%K A356936 nonn
%O A356936 1,6
%A A356936 _Gus Wiseman_, Sep 08 2022