A386606 - OEIS

%I #74 Feb 26 2026 20:26:49

%S 105,165,195,210,231,255,273,285,315,330,345,357,385,390,399,420,429,

%T 435,455,462,465,483,495,510,525,546,555,561,570,585,595,609,615,627,

%U 630,645,651,660,663,665,690,693,705,714,715,735,741,759,765,770,777,780

%N Numbers with exactly 3 distinct odd prime factors.

%C The cyclotomic polynomials Phi_a(n) (see A013595 and A013596, which are the same for n >= 4) of these numbers may contain coefficients as given in the irregular triangle A013595(a(n),m) (or A013596(a(n),m)), which are equal to 2 or -2. It seems that most of these numbers satisfy this condition. Counterexamples begin a(5) = 231, a(15) = 399, a(18) = 435 (see also A391069).

%C Some of the cyclotomic polynomials of these numbers contain coefficients which are in {-3, 3}; for example for a(n) = 1309, 1330 and 1463 with n = 119, 123 and 139 respectively, or even coefficients in {-4, 4} as for example for a(297) = 2431, or in {-5, 5} as in a(353) = 2717.

%H Robert Israel, <a href="/A386606/b386606.txt">Table of n, a(n) for n = 1..10000</a>

%F Equals Union_{k >= 0} 2^k*A278569; i.e., A278569 is the subsequence of all odd terms in this sequence.

%e 390 is a term since 390 = 2 * 3 * 5 * 13 has exactly 3 odd prime factors (3,5,13).

%e 525 is a term since 525 = 3 * 5^2 * 7 has exactly 3 odd prime factors (3,5,7).

%p filter:= proc(n) local L;

%p   L:= ifactors(n)[2];

%p   nops(select(t -> t[1] <> 2, L)) = 3

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Dec 01 2025

%t Select[Range[800], PrimeNu[# / 2^IntegerExponent[#, 2]] == 3 &] (* _Amiram Eldar_, Nov 21 2025 *)

%o (PARI) is(k) = (omega(k >> valuation(k, 2))==3); \\ _Jason Yuen_, Dec 24 2025

%Y Cf. A013595, A013596, A065091, A278569, A391069.

%K nonn

%O 1,1

%A _A.H.M. Smeets_, Nov 21 2025