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A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

%I A356955 #6 Sep 13 2022 09:36:14
%S A356955 1,2,3,4,6,7,8,9,12,13,14,16,18,19,21,24,26,27,28,32,36,37,38,39,42,
%T A356955 48,49,52,53,54,56,57,61,63,64,72,74,76,78,81,84,89,91,96,98,104,106,
%U A356955 108,111,112,113,114,117,122,126,128,131,133,144,147,148,151,152
%N A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.
%C A356955 An initial interval is a set {1,2,...,n}  for some n >= 0.
%C A356955 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 A356955 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 A356955 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 A356955 The initial terms and corresponding multisets of multisets:
%e A356955    1: {}
%e A356955    2: {{}}
%e A356955    3: {{1}}
%e A356955    4: {{},{}}
%e A356955    6: {{},{1}}
%e A356955    7: {{1,1}}
%e A356955    8: {{},{},{}}
%e A356955    9: {{1},{1}}
%e A356955   12: {{},{},{1}}
%e A356955   13: {{1,2}}
%e A356955   14: {{},{1,1}}
%e A356955   16: {{},{},{},{}}
%e A356955   18: {{},{1},{1}}
%e A356955   19: {{1,1,1}}
%e A356955   21: {{1},{1,1}}
%e A356955   24: {{},{},{},{1}}
%e A356955   26: {{},{1,2}}
%e A356955   27: {{1},{1},{1}}
%e A356955   28: {{},{},{1,1}}
%e A356955   32: {{},{},{},{},{}}
%t A356955 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356955 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A356955 Select[Range[100],And@@normQ/@primeMS/@primeMS[#]&]
%Y A356955 Multisets covering an initial interval are ctd by A011782, rkd by A055932.
%Y A356955 This is the initial version of A356944.
%Y A356955 Other types: A034691, A089259, A356945, A356954.
%Y A356955 Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
%Y A356955 A000041 counts integer partitions, strict A000009.
%Y A356955 A000670 counts patterns, ranked by A333217, necklace A019536.
%Y A356955 A000688 counts factorizations into prime powers.
%Y A356955 A001055 counts factorizations.
%Y A356955 A001221 counts prime divisors, sum A001414.
%Y A356955 A001222 counts prime factors with multiplicity.
%Y A356955 A056239 adds up prime indices, row sums of A112798.
%Y A356955 Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969, A349050, A349055.
%K A356955 nonn
%O A356955 1,2
%A A356955 _Gus Wiseman_, Sep 12 2022