A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

4, 24, 36, 54, 81, 160, 240, 360, 400, 540, 600, 810, 896, 900, 1000, 1215, 1344, 1350, 1500, 2016, 2025, 2240, 2250, 2500, 3024, 3136, 3360, 3375, 3750, 4536, 4704, 5040, 5600, 5625, 5632, 6250, 6804, 7056, 7560, 7840, 8400, 8448, 9375, 10206, 10584, 10976

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..46.

Formula

Disjoint from A361909.

Example

The terms together with their prime indices begin:
      4: {1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     54: {1,2,2,2}
     81: {2,2,2,2}
    160: {1,1,1,1,1,3}
    240: {1,1,1,1,2,3}
    360: {1,1,1,2,2,3}
    400: {1,1,1,1,3,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    810: {1,2,2,2,2,3}
    896: {1,1,1,1,1,1,1,4}
    900: {1,1,2,2,3,3}
   1000: {1,1,1,3,3,3}
   1215: {2,2,2,2,2,3}
   1344: {1,1,1,1,1,1,2,4}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   2016: {1,1,1,1,1,2,2,4}
   2025: {2,2,2,2,3,3}
   2240: {1,1,1,1,1,1,3,4}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Length[prix[#]]==2*Max[prix[#]]&]

Crossrefs

The LHS (number of prime indices) is A001222.

The RHS is twice A061395.

Before multiplying by 2 we had A106529.

Partitions of this type are counted by A237753.

For sum instead of length we have A344415, counted by A035363.

An adjoint version is A361909, also counted by A237753.

For minimum instead of maximum we have A363134, counted by A237757.

A112798 lists prime indices, sum A056239.

A326567/A326568 gives mean of prime indices.

Cf. A001221, A067801, A118096, A324521, A237752, A361855, A361908, A363222.

Sequence in context: A333655 A189228 A338786 * A048188 A121024 A120622

Adjacent sequences: A363215 A363216 A363217 * A363219 A363220 A363221

Keyword

nonn

Author

Gus Wiseman, May 23 2023

Status

approved