A387119 Numbers whose prime indices all have exactly 2 divisors in common.

3, 5, 9, 11, 17, 21, 25, 27, 31, 39, 41, 57, 59, 63, 65, 67, 81, 83, 87, 91, 109, 111, 115, 117, 121, 125, 127, 129, 147, 157, 159, 171, 179, 183, 185, 189, 191, 203, 211, 213, 235, 237, 241, 243, 247, 261, 267, 273, 277, 283, 289, 299, 301, 303, 305, 319, 321

Offset: 1

Author

Gus Wiseman, Aug 21 2025

Keywords

• nonn
• new

Comments

All terms are odd.

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 87 are {2,10}, with divisors {{1,2},{1,2,5,10}}, with intersection {1,2}, so 87 is in the sequence.
The prime indices of 91 are {4,6}, with divisors {{1,2,4},{1,2,3,6}}, with intersection {1,2}, so 91 is in the sequence.
The terms together with their prime indices begin:
    3: {2}
    5: {3}
    9: {2,2}
   11: {5}
   17: {7}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   31: {11}
   39: {2,6}
   41: {13}
   57: {2,8}
   59: {17}
   63: {2,2,4}
   65: {3,6}
   67: {19}
   81: {2,2,2,2}
		

Crossrefs

For initial intervals instead of divisors we have A016945.

Positions of 1 are A289509, complement A318978.

Positions of 2 in A387114, for prime factors or indices A387135.

A000005 counts divisors.

A001414 adds up distinct prime divisors, counted by A001221.

A003963 multiplies together the prime indices of n.

A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.

A120383 lists numbers divisible by all of their prime indices.

A289508 gives greatest common divisor of prime indices.

Cf. A000720, A055396, A061395, A335433, A355731, A355733, A355739, A355740, A370820.

Programs

Mathematica

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