A371288 Numbers whose distinct prime indices form the set of divisors of some positive integer.

2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 32, 34, 36, 40, 42, 44, 48, 50, 54, 62, 64, 68, 72, 80, 82, 84, 88, 96, 100, 108, 118, 124, 126, 128, 134, 136, 144, 160, 162, 164, 166, 168, 176, 192, 200, 216, 218, 230, 236, 242, 248, 250, 252, 254, 256, 268, 272, 288

Offset

1,1

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..59.

Example

The prime indices of 694782 are {1,2,2,5,5,5,10} with distinct elements {1,2,5,10}, which form the set of divisors of 10, so 694782 is in the sequence.
The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Mathematica

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

Crossrefs

The squarefree case is A371283, unsorted version A275700.

Partitions of this type are counted by A371284, strict A054973.

Products of squarefree terms are A371286, unsorted version A371285.

A000005 counts divisors.

A001221 counts distinct prime factors.

A027746 lists prime factors, indices A112798, length A001222.

Cf. A000720, A003963, A005179, A007416, A034729, A370820, A371131, A371177.

Sequence in context: A371177 A076450 A097379 * A114871 A085150 A051178

Adjacent sequences: A371285 A371286 A371287 * A371289 A371290 A371291

Keyword

nonn

Author

Gus Wiseman, Mar 22 2024

Status

approved