A356940 - OEIS

%I #6 Sep 13 2022 09:36:59

%S 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 104,108,113,117,128,144,156,162,169,192,208,216,226,234,243,256,288,

%U 312,324,338,339,351,384,416,432,452,468,486,507,512,576,624,648

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

%C An initial interval is a set {1,2,...,n}  for some n >= 0.

%C 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 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 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 The initial terms and corresponding multisets of multisets:

%e    1: {}

%e    2: {{}}

%e    3: {{1}}

%e    4: {{},{}}

%e    6: {{},{1}}

%e    8: {{},{},{}}

%e    9: {{1},{1}}

%e   12: {{},{},{1}}

%e   13: {{1,2}}

%e   16: {{},{},{},{}}

%e   18: {{},{1},{1}}

%e   24: {{},{},{},{1}}

%e   26: {{},{1,2}}

%e   27: {{1},{1},{1}}

%e   32: {{},{},{},{},{}}

%e   36: {{},{},{1},{1}}

%e   39: {{1},{1,2}}

%e   48: {{},{},{},{},{1}}

%e   52: {{},{},{1,2}}

%e   54: {{},{1},{1},{1}}

%e   64: {{},{},{},{},{},{}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t chinQ[y_]:=y==Range[Length[y]];

%t Select[Range[100],And@@chinQ/@primeMS/@primeMS[#]&]

%Y This is the initial version of A356939.

%Y Initial intervals are counted by A010054, ranked by A002110.

%Y Other types: A007294, A322585.

%Y Other conditions: A302478, A302492, A356930, A356935, A356944, A356955.

%Y A000041 counts integer partitions, strict A000009.

%Y A000688 counts factorizations into prime powers.

%Y A001055 counts factorizations.

%Y A001221 counts prime divisors, sum A001414.

%Y A001222 counts prime factors with multiplicity.

%Y A056239 adds up prime indices, row sums of A112798.

%Y Cf. A000040, A000720, A003963, A055887, A076610, A302242, A304969.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 12 2022