A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 55, 59, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 97, 99, 101, 105, 109, 115, 117, 119, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 147, 149

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}}.

Links

Table of n, a(n) for n=1..61.

Example

The terms together with the corresponding multigraphs begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[#]&&Max@@Length/@prix/@prix[#]<=2&]

Crossrefs

In the unlabeled case these multigraphs are counted by A320663.

These are products of primes indexed by elements of A037143 greater than 1.

For just primes we have A076610, squarefree A302590.

For just semiprimes we have A339112, squarefree A340020.

For just half-loops we have A340019.

This is the odd case of A368728, complement A368833.

A000607 counts partitions into primes, with ones allowed A034891.

A001358 lists semiprimes, squarefree A006881.

A006450, A106349, A322551, A368732 list selected primes.

A056239 adds up prime indices, row sums of A112798.

A101048 counts partitions into semiprimes.

Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320912, A339113.

Sequence in context: A248608 A362680 A309325 * A192863 A143450 A225563

Adjacent sequences: A368726 A368727 A368728 * A368730 A368731 A368732

Keyword

nonn

Author

Gus Wiseman, Jan 07 2024

Status

approved