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

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