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Positive integers k such that 2*omega(k) <= bigomega(k).
6

%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