A368603 Products of odd primes of squarefree index. MM-numbers of set multipartitions.

1, 3, 5, 9, 11, 13, 15, 17, 25, 27, 29, 31, 33, 39, 41, 43, 45, 47, 51, 55, 59, 65, 67, 73, 75, 79, 81, 83, 85, 87, 93, 99, 101, 109, 113, 117, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 149, 153, 155, 157, 163, 165, 167, 169, 177, 179, 181, 187

Offset

1,2

Comments

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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of 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}}.

A set multipartition is a finite multiset of finite nonempty sets.

Links

Table of n, a(n) for n=1..59.

Example

The terms together with the corresponding set multipartitions begin:
   1: {}
   3: {{1}}
   5: {{2}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[#]&&And@@SquareFreeQ/@prix[#]&]

Crossrefs

Odd case of A302478.

Products of odd terms of A302491.

A049311 counts non-isomorphic set multipartitions, strict A283877.

A050320 counts set multipartitions of prime indices.

A056239 adds up prime indices, row sums of A112798.

A089259 counts set multipartitions of integer partitions.

A116540 counts set multipartitions covering an initial interval by weight.

A368533 lists numbers with squarefree binary indices.

Cf. A000040, A000720, A001222, A005117, A006450, A076610, A270995, A296119, A302242, A302590, A339113.

Sequence in context: A296243 A162699 A325129 * A173263 A285141 A329095

Adjacent sequences: A368600 A368601 A368602 * A368604 A368605 A368606

Keyword

nonn

Author

Gus Wiseman, Jan 08 2024

Status

approved