cp's OEIS Frontend

A361393 Positive integers k such that 2*omega(k) > bigomega(k).

%I A361393 #10 Mar 21 2023 07:01:45
%S A361393 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 A361393 34,35,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,62,
%U A361393 63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84,85
%N A361393 Positive integers k such that 2*omega(k) > bigomega(k).
%C A361393 First differs from A317090 in having 120 and lacking 360.
%C A361393 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 A361393 {k: 2*A001221(k) > A001222(k)}. - _R. J. Mathar_, Mar 21 2023
%e A361393 The terms together with their prime indices begin:
%e A361393     2: {1}
%e A361393     3: {2}
%e A361393     5: {3}
%e A361393     6: {1,2}
%e A361393     7: {4}
%e A361393    10: {1,3}
%e A361393    11: {5}
%e A361393    12: {1,1,2}
%e A361393    13: {6}
%e A361393    14: {1,4}
%e A361393    15: {2,3}
%e A361393    17: {7}
%e A361393    18: {1,2,2}
%e A361393    19: {8}
%e A361393    20: {1,1,3}
%e A361393 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 A361393 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 A361393 isA361393 := proc(n)
%p A361393     if 2*A001221(n) > numtheory[bigomega](n) then
%p A361393         true;
%p A361393     else
%p A361393         false ;
%p A361393     end if:
%p A361393 end proc:
%p A361393 for n from 1 to 100 do
%p A361393     if isA361393(n) then
%p A361393         printf("%d,",n) ;
%p A361393     end if;
%p A361393 end do: # _R. J. Mathar_, Mar 21 2023
%t A361393 Select[Range[1000],2*PrimeNu[#]>PrimeOmega[#]&]
%Y A361393 These partitions are counted by A237365.
%Y A361393 The complement is A361204.
%Y A361393 A001221 (omega) counts distinct prime factors.
%Y A361393 A001222 (bigomega) counts prime factors.
%Y A361393 A112798 lists prime indices, sum A056239.
%Y A361393 A326567/A326568 gives mean of prime indices.
%Y A361393 A360005 gives median of prime indices (times 2), distinct A360457.
%Y A361393 Comparing twice the number of distinct parts to the number of parts:
%Y A361393               less: A360254, ranks A360558
%Y A361393              equal: A239959, ranks A067801
%Y A361393            greater: A237365, ranks A361393
%Y A361393      less or equal: A237363, ranks A361204
%Y A361393   greater or equal: A361394, ranks A361395
%Y A361393 Cf. A046660, A061395, A067340, A111907.
%Y A361393 Cf. A324515, A324517, A324519, A324521, A324522, A324560, A324562.
%K A361393 nonn
%O A361393 1,1
%A A361393 _Gus Wiseman_, Mar 16 2023