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

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, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Author

Gus Wiseman, Sep 12 2022

Keywords

• nonn

Comments

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.

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}}.

Examples

			The initial terms and corresponding multiset partitions:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}
		

Links

• Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Crossrefs

Gapless multisets are counted by A034296, ranked by A073491.

The initial version is A356955.

Other types: A356233, A356941, A356942, A356943.

Other conditions: A302478, A302492, A356930, A356935, A356939, A356940.

A000041 counts integer partitions, strict A000009.

A000688 counts factorizations into prime powers.

A001055 counts factorizations.

A001221 counts prime divisors, sum A001414.

A001222 counts prime factors with multiplicity.

A011782 counts multisets covering an initial interval.

A056239 adds up prime indices, row sums of A112798.

Cf. A000720, A003963, A076610, A270995, A302242, A349050, A349055.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
Select[Range[100],And@@nogapQ/@primeMS/@primeMS[#]&]