A360558 Numbers whose multiset of prime factors (or indices, see A112798) has more adjacent equalities (or parts that have appeared before) than distinct parts.

8, 16, 27, 32, 48, 64, 72, 80, 81, 96, 108, 112, 125, 128, 144, 160, 162, 176, 192, 200, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 343, 352, 368, 384, 392, 400, 405, 416, 432, 448, 464, 480, 486, 496, 500, 512, 544, 567, 576, 592, 608, 625, 640, 648

Offset

1,1

Comments

No terms are squarefree.

Also numbers whose first differences of 0-prepended prime indices have median 0.

Links

Table of n, a(n) for n=1..54.

Formula

A001222(a(n)) > 2*A001221(a(n)).

Example

The terms together with their prime indices begin:
     8: {1,1,1}
    16: {1,1,1,1}
    27: {2,2,2}
    32: {1,1,1,1,1}
    48: {1,1,1,1,2}
    64: {1,1,1,1,1,1}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}
    81: {2,2,2,2}
    96: {1,1,1,1,1,2}
   108: {1,1,2,2,2}
   112: {1,1,1,1,4}
   125: {3,3,3}
For example, the prime indices of 720 are {1,1,1,1,2,2,3} with 4 adjacent equalities and 3 distinct parts, so 720 is in the sequence.

Mathematica

Select[Range[100], PrimeOmega[#]>2*PrimeNu[#]&]

Crossrefs

For equality we have A067801.

These partitions are counted by A360254.

A112798 lists prime indices, length A001222, sum A056239.

A326567/A326568 gives mean of prime indices.

A360005 gives median of prime indices (times 2).

Cf. A000975, A027193, A067340, A237363, A317090, A360248, A360249, A360454, A360555, A360556.

Sequence in context: A329134 A090081 A059172 * A107606 A354561 A349306

Adjacent sequences: A360555 A360556 A360557 * A360559 A360560 A360561

Keyword

nonn

Author

Gus Wiseman, Feb 20 2023

Status

approved