A356955 MM-numbers of multisets of multisets, each covering an initial interval. Products of primes indexed by elements of A055932.

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 16, 18, 19, 21, 24, 26, 27, 28, 32, 36, 37, 38, 39, 42, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 72, 74, 76, 78, 81, 84, 89, 91, 96, 98, 104, 106, 108, 111, 112, 113, 114, 117, 122, 126, 128, 131, 133, 144, 147, 148, 151, 152

Offset

1,2

Comments

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

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..63.

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: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  32: {{},{},{},{},{}}

Mathematica

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

Crossrefs

Multisets covering an initial interval are ctd by A011782, rkd by A055932.

This is the initial version of A356944.

Other types: A034691, A089259, A356945, A356954.

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

A000041 counts integer partitions, strict A000009.

A000670 counts patterns, ranked by A333217, necklace A019536.

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, A349050, A349055.

Sequence in context: A236207 A225837 A035246 * A112258 A032893 A032852

Adjacent sequences: A356952 A356953 A356954 * A356956 A356957 A356958

Keyword

nonn

Author

Gus Wiseman, Sep 12 2022

Status

approved