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A356944 MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.

%I A356944 #7 Sep 13 2022 09:36:37
%S A356944 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A356944 27,28,30,31,32,33,34,35,36,37,38,39,40,41,42,44,45,46,47,48,49,50,51,
%U A356944 52,53,54,55,56,57,59,60,61,62,63,64,65,66,67,68,69,70
%N A356944 MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.
%C A356944 A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.
%C A356944 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 A356944 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 A356944 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 A356944 The initial terms and corresponding multiset partitions:
%e A356944    1: {}
%e A356944    2: {{}}
%e A356944    3: {{1}}
%e A356944    4: {{},{}}
%e A356944    5: {{2}}
%e A356944    6: {{},{1}}
%e A356944    7: {{1,1}}
%e A356944    8: {{},{},{}}
%e A356944    9: {{1},{1}}
%e A356944   10: {{},{2}}
%e A356944   11: {{3}}
%e A356944   12: {{},{},{1}}
%e A356944   13: {{1,2}}
%e A356944   14: {{},{1,1}}
%e A356944   15: {{1},{2}}
%e A356944   16: {{},{},{},{}}
%t A356944 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356944 nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
%t A356944 Select[Range[100],And@@nogapQ/@primeMS/@primeMS[#]&]
%Y A356944 Gapless multisets are counted by A034296, ranked by A073491.
%Y A356944 The initial version is A356955.
%Y A356944 Other types: A356233, A356941, A356942, A356943.
%Y A356944 Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.
%Y A356944 A000041 counts integer partitions, strict A000009.
%Y A356944 A000688 counts factorizations into prime powers.
%Y A356944 A001055 counts factorizations.
%Y A356944 A001221 counts prime divisors, sum A001414.
%Y A356944 A001222 counts prime factors with multiplicity.
%Y A356944 A011782 counts multisets covering an initial interval.
%Y A356944 A056239 adds up prime indices, row sums of A112798.
%Y A356944 Cf. A000720, A003963, A076610, A270995, A302242, A349050, A349055.
%K A356944 nonn
%O A356944 1,2
%A A356944 _Gus Wiseman_, Sep 12 2022