A368101 Numbers of which there is exactly one way to choose a different prime factor of each prime index.

1, 3, 5, 11, 15, 17, 31, 33, 39, 41, 51, 55, 59, 65, 67, 83, 85, 87, 93, 109, 111, 123, 127, 129, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 213, 235, 237, 241, 249, 255, 267, 277, 283, 295, 303, 305, 319, 321, 327, 331, 335, 341, 353, 365, 367, 381

Offset: 1

Author

Gus Wiseman, Dec 12 2023

Keywords

• nonn

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.

Examples

			The prime indices of 2795 are {3,6,14}, with prime factors {{3},{2,3},{2,7}}, and the only choice with different terms is {3,2,7}, so 2795 is in the sequence.
The terms together with their prime indices of prime indices begin:
    1: {}
    3: {{1}}
    5: {{2}}
   11: {{3}}
   15: {{1},{2}}
   17: {{4}}
   31: {{5}}
   33: {{1},{3}}
   39: {{1},{1,2}}
   41: {{6}}
   51: {{1},{4}}
   55: {{2},{3}}
   59: {{7}}
   65: {{2},{1,2}}
   67: {{8}}
   83: {{9}}
   85: {{2},{4}}
   87: {{1},{1,3}}
   93: {{1},{5}}
  109: {{10}}
  111: {{1},{1,1,2}}
		

Crossrefs

For no choices we have A355529, odd A355535, binary A367907.

Positions of ones in A367771.

The version for binary indices is A367908, positions of ones in A367905.

For any number of choices we have A368100.

For a unique set instead of sequence we have A370647, counted by A370594.

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

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

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

A355741 chooses a prime factor of each prime index, multisets A355744.

Cf. A007716, A355737, A355739, A355740, A355745, A367904, A367906, A370584.

Programs

Mathematica

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