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Number of multiset partitions of the multiset of prime indices of n into intervals. Number of factorizations of n into members of A073485.
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%I #8 Sep 10 2022 07:35:28

%S 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 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 1,2,1,3,1,1,2,1,2,2,1,1,1,1,1,2,1,1,1

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

%C An interval is a set of positive integers with all differences of adjacent elements equal to 1.

%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 = 6, 30, 36, 90, 180:

%e {12} {123} {12}{12} {12}{23} {12}{123}

%e {1}{2} {1}{23} {1}{2}{12} {2}{123} {1}{12}{23}

%e {3}{12} {1}{1}{2}{2} {1}{2}{23} {1}{2}{123}

%e {1}{2}{3} {2}{3}{12} {3}{12}{12}

%e {1}{2}{2}{3} {1}{1}{2}{23}

%e {1}{2}{3}{12}

%e {1}{1}{2}{2}{3}

%e The a(n) factorizations for n = 6, 30, 36, 90, 180:

%e (6) (30) (6*6) (3*30) (6*30)

%e (2*3) (5*6) (2*3*6) (6*15) (5*6*6)

%e (2*15) (2*2*3*3) (3*5*6) (2*3*30)

%e (2*3*5) (2*3*15) (2*6*15)

%e (2*3*3*5) (2*3*5*6)

%e (2*2*3*15)

%e (2*2*3*3*5)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

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

%t chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];

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

%Y A000688 counts factorizations into prime powers.

%Y A001055 counts factorizations.

%Y A001221 counts prime divisors, sum A001414.

%Y A001222 counts prime factors with multiplicity.

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

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

%Y Intervals are counted by A000012, A001227, ranked by A073485.

%Y Other types: A107742, A356233, A356937, A356938, A356939.

%Y Other conditions: A050320, A050330, A322585, A356931, A356945.

%Y Cf. A003963, A073491, A287170, A328195, A356234.

%K nonn

%O 1,6

%A _Gus Wiseman_, Sep 08 2022