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A356930 Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.

%I A356930 #11 Sep 12 2022 22:48:48
%S A356930 1,2,3,4,6,7,8,9,11,12,14,16,18,19,21,22,24,27,28,29,31,32,33,36,38,
%T A356930 42,44,48,49,53,54,56,57,58,59,62,63,64,66,71,72,76,77,79,81,83,84,87,
%U A356930 88,93,96,97,98,99,106,108,112,114,116,118,121,124,126,127
%N A356930 Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.
%C A356930 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 combined 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 A356930 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 A356930 The initial terms and corresponding multisets of multisets:
%e A356930    1: {}
%e A356930    2: {{}}
%e A356930    3: {{1}}
%e A356930    4: {{},{}}
%e A356930    6: {{},{1}}
%e A356930    7: {{1,1}}
%e A356930    8: {{},{},{}}
%e A356930    9: {{1},{1}}
%e A356930   11: {{3}}
%e A356930   12: {{},{},{1}}
%e A356930   14: {{},{1,1}}
%e A356930   16: {{},{},{},{}}
%e A356930   18: {{},{1},{1}}
%e A356930   19: {{1,1,1}}
%e A356930   21: {{1},{1,1}}
%e A356930   22: {{},{3}}
%e A356930   24: {{},{},{},{1}}
%e A356930   27: {{1},{1},{1}}
%e A356930   28: {{},{},{1,1}}
%e A356930   29: {{1,3}}
%e A356930   31: {{5}}
%e A356930   32: {{},{},{},{},{}}
%t A356930 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356930 Select[Range[100],And@@(OddQ[Times@@primeMS[#]]&/@primeMS[#])&]
%Y A356930 Multisets of odd numbers are counted by A000009, ranked by A066208.
%Y A356930 Factorizations of this type are counted by A356931.
%Y A356930 The version for odd lengths instead of parts is A356935, ranked by A089259.
%Y A356930 Other conditions: A302478, A302492, A356939, A356940, A356944, A356955.
%Y A356930 A000041 counts integer partitions, strict A000009.
%Y A356930 A000688 counts factorizations into prime powers.
%Y A356930 A001055 counts factorizations.
%Y A356930 A001221 counts prime divisors, sum A001414.
%Y A356930 A001222 counts prime factors with multiplicity.
%Y A356930 A056239 adds up prime indices, row sums of A112798.
%Y A356930 Cf. A000720, A003963, A011782, A055887, A076610, A302242, A304969.
%K A356930 nonn
%O A356930 1,2
%A A356930 _Gus Wiseman_, Sep 11 2022