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A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).

%I A356940 #6 Sep 13 2022 09:36:59
%S A356940 1,2,3,4,6,8,9,12,13,16,18,24,26,27,32,36,39,48,52,54,64,72,78,81,96,
%T A356940 104,108,113,117,128,144,156,162,169,192,208,216,226,234,243,256,288,
%U A356940 312,324,338,339,351,384,416,432,452,468,486,507,512,576,624,648
%N A356940 MM-numbers of multisets of initial intervals. Products of elements of A062447 (primes indexed by primorials A002110).
%C A356940 An initial interval is a set {1,2,...,n}  for some n >= 0.
%C A356940 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 A356940 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 A356940 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 A356940 The initial terms and corresponding multisets of multisets:
%e A356940    1: {}
%e A356940    2: {{}}
%e A356940    3: {{1}}
%e A356940    4: {{},{}}
%e A356940    6: {{},{1}}
%e A356940    8: {{},{},{}}
%e A356940    9: {{1},{1}}
%e A356940   12: {{},{},{1}}
%e A356940   13: {{1,2}}
%e A356940   16: {{},{},{},{}}
%e A356940   18: {{},{1},{1}}
%e A356940   24: {{},{},{},{1}}
%e A356940   26: {{},{1,2}}
%e A356940   27: {{1},{1},{1}}
%e A356940   32: {{},{},{},{},{}}
%e A356940   36: {{},{},{1},{1}}
%e A356940   39: {{1},{1,2}}
%e A356940   48: {{},{},{},{},{1}}
%e A356940   52: {{},{},{1,2}}
%e A356940   54: {{},{1},{1},{1}}
%e A356940   64: {{},{},{},{},{},{}}
%t A356940 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356940 chinQ[y_]:=y==Range[Length[y]];
%t A356940 Select[Range[100],And@@chinQ/@primeMS/@primeMS[#]&]
%Y A356940 This is the initial version of A356939.
%Y A356940 Initial intervals are counted by A010054, ranked by A002110.
%Y A356940 Other types: A007294, A322585.
%Y A356940 Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.
%Y A356940 A000041 counts integer partitions, strict A000009.
%Y A356940 A000688 counts factorizations into prime powers.
%Y A356940 A001055 counts factorizations.
%Y A356940 A001221 counts prime divisors, sum A001414.
%Y A356940 A001222 counts prime factors with multiplicity.
%Y A356940 A056239 adds up prime indices, row sums of A112798.
%Y A356940 Cf. A000040, A000720, A003963, A055887, A076610, A302242, A304969.
%K A356940 nonn
%O A356940 1,2
%A A356940 _Gus Wiseman_, Sep 12 2022