A377713 Squarefree composite k such that floor(log n/log lpf(k)) <= omega(k), where lpf = A020639 and omega = A001221.

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 385, 391, 403, 407, 437, 451, 455, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 595, 611, 629, 649, 667, 671, 689

Offset

1,1

Comments

Also squarefree composite k such that there exist no numbers m such that rad(m) | k and omega(m) > omega(k).

The only even term is 6.

Let P(i) = A002110(i). Numbers k = prime(i) * P(i+j)/P(i) < prime(i)^(i+j) with j ≥ 1 implies k such that omega(k) = j+1 is in the sequence.

The number k = p*m is a solution where squarefree m with lpf(m) > p is such that m < p^omega(m). For example, k = 5*7 is in the sequence since 7 < 5^2.

The number of a(n) such that lpf(a(n)) = p is finite. For example, the only terms divisible by 3 are {6, 15, 21}.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024.

Example

6 is in the sequence since floor(log_2 6) = 1+floor(log_2 3) = omega(6) = 2.
10 is not in the sequence since floor(log_2 5) = 2 and omega(10) = 2, thus 1+floor(log_2 5) > omega(10). Seen another way, 2^3 < 10, but omega(8) > omega(10).
15 is in the sequence since floor(log_3 15) = 1+floor(log_3 5) = omega(15) = 2.
21 is in the sequence because 1+floor(log_3 7) = omega(21) = 2.
33 = 3*11 is not in the sequence because 11 > 3^2.
115 = 5*23 is in the sequence because 23 < 5^2.
145 = 5*29 is not in the sequence since 29 > 5^2, etc.

Mathematica

s = Select[Range[1000], And[SquarefreeQ[#], CompositeQ[#]] &];
Select[s, Floor@ Log[FactorInteger[#][[1, 1]], #] <= PrimeOmega[#] &]

Crossrefs

Cf. A000961, A001221, A002110, A007947, A020639, A120944, A376846, A377590, A377591.

Sequence in context: A373899 A261078 A063466 * A138109 A377792 A332877

Adjacent sequences: A377710 A377711 A377712 * A377714 A377715 A377716

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 04 2024

Status

approved