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

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Author

Gus Wiseman, Mar 16 2023

Keywords

• nonn

Comments

First differs from A317090 in having 120 and lacking 360.

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

Examples

			The terms together with their prime indices begin:
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   12: {1,1,2}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
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.
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.
		

Crossrefs

These partitions are counted by A237365.

The complement is A361204.

A001221 (omega) counts distinct prime factors.

A001222 (bigomega) counts prime factors.

A112798 lists prime indices, sum A056239.

A326567/A326568 gives mean of prime indices.

A360005 gives median of prime indices (times 2), distinct A360457.

Comparing twice the number of distinct parts to the number of parts:

less: A360254, ranks A360558

equal: A239959, ranks A067801

greater: A237365, ranks A361393

less or equal: A237363, ranks A361204

greater or equal: A361394, ranks A361395

Cf. A046660, A061395, A067340, A111907.

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

Programs

Maple

isA361393 := proc(n)
    if 2*A001221(n) > numtheory[bigomega](n) then
        true;
    else
        false ;
    end if:
end proc:
for n from 1 to 100 do
    if isA361393(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Mar 21 2023

Mathematica

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

Formula

{k: 2*A001221(k) > A001222(k)}. - R. J. Mathar, Mar 21 2023