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

Author

Gus Wiseman, Feb 20 2023

Keywords

• nonn

Comments

No terms are squarefree.

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

Examples

			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.
		

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.

Programs

Mathematica

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

Formula

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