A322554 Numbers whose product of prime indices is a power of a squarefree number (A072774).

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, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80

Offset

1,2

Comments

The complement is {35, 37, 39, 45, 61, 65, ...}.

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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of regular multiset multisystems, where regularity means all vertex-degrees are equal.

Links

Table of n, a(n) for n=1..67.

Example

Most small numbers belong to this sequence. However, the sequence of multiset multisystems whose MM-numbers do not belong to this sequence begins:
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  45: {{1},{1},{2}}
  61: {{1,2,2}}
  65: {{2},{1,2}}
  69: {{1},{2,2}}
  70: {{},{2},{1,1}}
  71: {{1,1,3}}
  74: {{},{1,1,2}}
  75: {{1},{2},{2}}
  77: {{1,1},{3}}
  78: {{},{1},{1,2}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  90: {{},{1},{1},{2}}
  91: {{1,1},{1,2}}
  95: {{2},{1,1,1}}
  99: {{1},{1},{3}}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], SameQ@@Last/@FactorInteger[Times@@primeMS[#]]&]

Crossrefs

Cf. A003963, A056239, A062503, A072774, A112798, A302242, A302491, A320325, A320698, A322553.

Sequence in context: A273883 A132145 A272284 * A330232 A273882 A277112

Adjacent sequences: A322551 A322552 A322553 * A322555 A322556 A322557

Keyword

nonn

Author

Gus Wiseman, Dec 15 2018

Status

approved