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A361204 Positive integers k such that 2*omega(k) <= bigomega(k). 6
1, 4, 8, 9, 16, 24, 25, 27, 32, 36, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 100, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
FORMULA
A001222(a(n)) >= 2*A001221(a(n)).
EXAMPLE
The terms together with their prime indices begin:
1: {}
4: {1,1}
8: {1,1,1}
9: {2,2}
16: {1,1,1,1}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
49: {4,4}
54: {1,2,2,2}
56: {1,1,1,4}
64: {1,1,1,1,1,1}
MAPLE
filter:= proc(n) local F, t;
F:= ifactors(n)[2];
add(t[2], t=F) >= 2*nops(F)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 22 2023
MATHEMATICA
Select[Range[100], 2*PrimeNu[#]<=PrimeOmega[#]&]
CROSSREFS
These partitions are counted by A237363.
The complement is A361393.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors.
A112798 lists prime indices, sum A056239.
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: A280387 A243180 A100657 * A245080 A212164 A293243
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 14 2023
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)