A377713 - OEIS

%I #6 Nov 13 2024 17:17:29

%S 6,15,21,35,55,65,77,85,91,95,115,119,133,143,161,187,203,209,217,221,

%T 247,253,259,287,299,301,319,323,329,341,377,385,391,403,407,437,451,

%U 455,473,481,493,517,527,533,551,559,583,589,595,611,629,649,667,671,689

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

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

%C The only even term is 6.

%C 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.

%C 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.

%C 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}.

%H Michael De Vlieger, <a href="/A377713/b377713.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.18645.64480">Numbers k for which floor(log k / log lpf(k)) <= bigomega(k)</a>, 2024.

%e 6 is in the sequence since floor(log_2 6) = 1+floor(log_2 3) = omega(6) = 2.

%e 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).

%e 15 is in the sequence since floor(log_3 15) = 1+floor(log_3 5) = omega(15) = 2.

%e 21 is in the sequence because 1+floor(log_3 7) = omega(21) = 2.

%e 33 = 3*11 is not in the sequence because 11 > 3^2.

%e 115 = 5*23 is in the sequence because 23 < 5^2.

%e 145 = 5*29 is not in the sequence since 29 > 5^2, etc.

%t s = Select[Range[1000], And[SquarefreeQ[#], CompositeQ[#]] &];

%t Select[s, Floor@ Log[FactorInteger[#][[1, 1]], #] <= PrimeOmega[#] &]

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

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Nov 04 2024