A356939 MM-numbers of multisets of intervals. Products of primes indexed by members of A073485.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 30, 31, 32, 33, 34, 36, 39, 40, 41, 44, 45, 47, 48, 50, 51, 52, 54, 55, 59, 60, 62, 64, 65, 66, 67, 68, 72, 75, 78, 80, 81, 82, 83, 85, 88, 90, 93, 94, 96, 99, 100, 102, 104, 108

Offset

1,2

Comments

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

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}}.

Links

Table of n, a(n) for n=1..66.

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

Example

The initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   6: {{},{1}}
   8: {{},{},{}}
   9: {{1},{1}}
  10: {{},{2}}
  11: {{3}}
  12: {{},{},{1}}
  13: {{1,2}}
  15: {{1},{2}}
  16: {{},{},{},{}}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
chQ[y_]:=Or[Length[y]<=1, Union[Differences[y]]=={1}];
Select[Range[100], And@@chQ/@primeMS/@primeMS[#]&]

Crossrefs

The initial version is A356940.

Intervals are counted by A000012, A001227, ranked by A073485.

Other types: A107742, A356936, A356937, A356938.

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

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.

A056239 adds up prime indices, row sums of A112798.

Cf. A000040, A000720, A003963, A055887, A076610, A270995, A302242, A304969.

Sequence in context: A332107 A267305 A302478 * A113619 A325114 A004765

Adjacent sequences: A356936 A356937 A356938 * A356940 A356941 A356942

Keyword

nonn

Author

Gus Wiseman, Sep 12 2022

Status

approved