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A361393
Positive integers k such that 2*omega(k) > bigomega(k).
6
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
1,1
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
FORMULA
{k: 2*A001221(k) > A001222(k)}. - R. J. Mathar, Mar 21 2023
EXAMPLE
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.
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[#]&]
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
Sequence in context: A212167 A339741 A317090 * A179983 A065872 A028741
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 16 2023
STATUS
approved