%I #13 Mar 23 2023 03:45:45
%S 1,4,8,9,16,24,25,27,32,36,40,48,49,54,56,64,72,80,81,88,96,100,104,
%T 108,112,121,125,128,135,136,144,152,160,162,169,176,184,189,192,196,
%U 200,208,216,224,225,232,240,243,248,250,256,272,288,289,296,297,304
%N Positive integers k such that 2*omega(k) <= bigomega(k).
%H Robert Israel, <a href="/A361204/b361204.txt">Table of n, a(n) for n = 1..10000</a>
%F A001222(a(n)) >= 2*A001221(a(n)).
%e The terms together with their prime indices begin:
%e 1: {}
%e 4: {1,1}
%e 8: {1,1,1}
%e 9: {2,2}
%e 16: {1,1,1,1}
%e 24: {1,1,1,2}
%e 25: {3,3}
%e 27: {2,2,2}
%e 32: {1,1,1,1,1}
%e 36: {1,1,2,2}
%e 40: {1,1,1,3}
%e 48: {1,1,1,1,2}
%e 49: {4,4}
%e 54: {1,2,2,2}
%e 56: {1,1,1,4}
%e 64: {1,1,1,1,1,1}
%p filter:= proc(n) local F,t;
%p F:= ifactors(n)[2];
%p add(t[2],t=F) >= 2*nops(F)
%p end proc:
%p select(filter, [$1..1000]); # _Robert Israel_, Mar 22 2023
%t Select[Range[100],2*PrimeNu[#]<=PrimeOmega[#]&]
%Y These partitions are counted by A237363.
%Y The complement is A361393.
%Y A001221 (omega) counts distinct prime factors.
%Y A001222 (bigomega) counts prime factors.
%Y A112798 lists prime indices, sum A056239.
%Y A360005 gives median of prime indices (times 2), distinct A360457.
%Y Comparing twice the number of distinct parts to the number of parts:
%Y less: A360254, ranks A360558
%Y equal: A239959, ranks A067801
%Y greater: A237365, ranks A361393
%Y less or equal: A237363, ranks A361204
%Y greater or equal: A361394, ranks A361395
%Y Cf. A046660, A061395, A067340, A111907.
%Y Cf. A324517, A324521, A324522, A324560, A324562.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 14 2023