A361204 Positive integers k such that 2*omega(k) <= bigomega(k).

1, 4, 8, 9, 16, 24, 25, 27, 32, 36, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 100, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304

Offset: 1

Author

Gus Wiseman, Mar 14 2023

Keywords

• nonn

Examples

			The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    48: {1,1,1,1,2}
    49: {4,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

These partitions are counted by A237363.

The complement is A361393.

A001221 (omega) counts distinct prime factors.

A001222 (bigomega) counts prime factors.

A112798 lists prime indices, sum A056239.

A360005 gives median of prime indices (times 2), distinct A360457.

Comparing twice the number of distinct parts to the number of parts:

less: A360254, ranks A360558

equal: A239959, ranks A067801

greater: A237365, ranks A361393

less or equal: A237363, ranks A361204

greater or equal: A361394, ranks A361395

Cf. A046660, A061395, A067340, A111907.

Cf. A324517, A324521, A324522, A324560, A324562.

Programs

Maple

filter:= proc(n) local F,t;
  F:= ifactors(n)[2];
  add(t[2],t=F) >= 2*nops(F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 22 2023

Mathematica

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

Formula

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