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

%I #10 Mar 21 2023 07:01:45

%S 2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,26,28,29,30,31,33,

%T 34,35,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,62,

%U 63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84,85

%N Positive integers k such that 2*omega(k) > bigomega(k).

%C First differs from A317090 in having 120 and lacking 360.

%C There are numbers like 1, 120, 168, 180, 252,... which are not in A179983 but in here, and others like 360, 504, 540, 600,... which are in A179983 but not in here. - _R. J. Mathar_, Mar 21 2023

%F {k: 2*A001221(k) > A001222(k)}. - _R. J. Mathar_, Mar 21 2023

%e The terms together with their prime indices begin:

%e 2: {1}

%e 3: {2}

%e 5: {3}

%e 6: {1,2}

%e 7: {4}

%e 10: {1,3}

%e 11: {5}

%e 12: {1,1,2}

%e 13: {6}

%e 14: {1,4}

%e 15: {2,3}

%e 17: {7}

%e 18: {1,2,2}

%e 19: {8}

%e 20: {1,1,3}

%e The prime indices of 120 are {1,1,1,2,3}, with 3 distinct parts and 5 parts, and 2*3 > 5, so 120 is in the sequence.

%e The prime indices of 360 are {1,1,1,2,2,3}, with 3 distinct parts and 6 parts, and 2*3 is not greater than 6, so 360 is not in the sequence.

%p isA361393 := proc(n)

%p if 2*A001221(n) > numtheory[bigomega](n) then

%p true;

%p else

%p false ;

%p end if:

%p end proc:

%p for n from 1 to 100 do

%p if isA361393(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Mar 21 2023

%t Select[Range[1000],2*PrimeNu[#]>PrimeOmega[#]&]

%Y These partitions are counted by A237365.

%Y The complement is A361204.

%Y A001221 (omega) counts distinct prime factors.

%Y A001222 (bigomega) counts prime factors.

%Y A112798 lists prime indices, sum A056239.

%Y A326567/A326568 gives mean of prime indices.

%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. A324515, A324517, A324519, A324521, A324522, A324560, A324562.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 16 2023