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A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.

%I A356935 #6 Sep 13 2022 09:37:16
%S A356935 1,3,5,9,11,15,17,19,25,27,31,33,37,41,45,51,55,57,59,61,67,71,75,81,
%T A356935 83,85,93,95,99,103,107,109,111,113,121,123,125,127,131,135,153,155,
%U A356935 157,165,171,177,179,181,183,185,187,191,193,197,201,205,209,211,213
%N A356935 Numbers whose prime indices all have odd bigomega (number of prime factors with multiplicity). Products of primes indexed by elements of A026424. MM-numbers of finite multisets of finite odd-length multisets of positive integers.
%C A356935 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. 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 A356935 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 A356935 The initial terms and corresponding multiset partitions:
%e A356935    1: {}
%e A356935    3: {{1}}
%e A356935    5: {{2}}
%e A356935    9: {{1},{1}}
%e A356935   11: {{3}}
%e A356935   15: {{1},{2}}
%e A356935   17: {{4}}
%e A356935   19: {{1,1,1}}
%e A356935   25: {{2},{2}}
%e A356935   27: {{1},{1},{1}}
%e A356935   31: {{5}}
%e A356935   33: {{1},{3}}
%e A356935   37: {{1,1,2}}
%e A356935   41: {{6}}
%e A356935   45: {{1},{1},{2}}
%e A356935   51: {{1},{4}}
%e A356935   55: {{2},{3}}
%e A356935   57: {{1},{1,1,1}}
%t A356935 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356935 Select[Range[100],OddQ[Times@@Length/@primeMS/@primeMS[#]]&]
%Y A356935 A000041 counts integer partitions, strict A000009.
%Y A356935 A000688 counts factorizations into prime powers.
%Y A356935 A001055 counts factorizations.
%Y A356935 A001221 counts prime divisors, sum A001414.
%Y A356935 A001222 counts prime factors with multiplicity.
%Y A356935 A056239 adds up prime indices, row sums of A112798.
%Y A356935 Odd-size multisets are ctd by A000302, A027193, A058695, rkd by A026424.
%Y A356935 Other types: A050330, A356932, A356933, A356934.
%Y A356935 Other conditions: A302478, A302492, A356930, A356939, A356940, A356944, A356945.
%Y A356935 Cf. A000040, A000720, A003963, A270995, A302242, A304969, A349050, A349055.
%K A356935 nonn
%O A356935 1,2
%A A356935 _Gus Wiseman_, Sep 12 2022