A368100 Numbers of which it is possible to choose a different prime factor of each prime index.

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 127, 129, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163

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.

Links

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

Example

The prime indices of 2849 are {4,5,12}, with prime factors {{2,2},{5},{2,2,3}}, and of the two choices (2,5,2) and (2,5,3) the latter has all different terms, so 2849 is in the sequence.
The terms together with their prime indices of prime indices begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}

Mathematica

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

Crossrefs

The complement is A355529, odd A355535, binary A367907.

Positions of positive terms in A367771.

The version for binary indices is A367906, positive positions in A367905.

For a unique choice we have A368101, binary A367908.

The version for divisors instead of factors is A368110, complement A355740.

A058891 counts set-systems, covering A003465, connected A323818.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

Cf. A092918, A355737, A355739, A355741, A355744, A355745, A367902.

Sequence in context: A272872 A359080 A103796 * A302568 A359260 A294674

Adjacent sequences: A368097 A368098 A368099 * A368101 A368102 A368103

Keyword

nonn

Author

Gus Wiseman, Dec 12 2023

Status

approved