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A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

%I A356939 #5 Sep 13 2022 09:37:04
%S A356939 1,2,3,4,5,6,8,9,10,11,12,13,15,16,17,18,20,22,24,25,26,27,30,31,32,
%T A356939 33,34,36,39,40,41,44,45,47,48,50,51,52,54,55,59,60,62,64,65,66,67,68,
%U A356939 72,75,78,80,81,82,83,85,88,90,93,94,96,99,100,102,104,108
%N A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.
%C A356939 An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.
%C A356939 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.
%C A356939 We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
%H A356939 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 A356939 The initial terms and corresponding multisets of multisets:
%e A356939    1: {}
%e A356939    2: {{}}
%e A356939    3: {{1}}
%e A356939    4: {{},{}}
%e A356939    5: {{2}}
%e A356939    6: {{},{1}}
%e A356939    8: {{},{},{}}
%e A356939    9: {{1},{1}}
%e A356939   10: {{},{2}}
%e A356939   11: {{3}}
%e A356939   12: {{},{},{1}}
%e A356939   13: {{1,2}}
%e A356939   15: {{1},{2}}
%e A356939   16: {{},{},{},{}}
%t A356939 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356939 chQ[y_]:=Or[Length[y]<=1,Union[Differences[y]]=={1}];
%t A356939 Select[Range[100],And@@chQ/@primeMS/@primeMS[#]&]
%Y A356939 The initial version is A356940.
%Y A356939 Intervals are counted by A000012, A001227, ranked by A073485.
%Y A356939 Other types: A107742, A356936, A356937, A356938.
%Y A356939 Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
%Y A356939 A000041 counts integer partitions, strict A000009.
%Y A356939 A000688 counts factorizations into prime powers.
%Y A356939 A001055 counts factorizations.
%Y A356939 A001221 counts prime divisors, sum A001414.
%Y A356939 A001222 counts prime factors with multiplicity.
%Y A356939 A056239 adds up prime indices, row sums of A112798.
%Y A356939 Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969.
%K A356939 nonn
%O A356939 1,2
%A A356939 _Gus Wiseman_, Sep 12 2022