A386606 Numbers with exactly 3 distinct odd prime factors.

105, 165, 195, 210, 231, 255, 273, 285, 315, 330, 345, 357, 385, 390, 399, 420, 429, 435, 455, 462, 465, 483, 495, 510, 525, 546, 555, 561, 570, 585, 595, 609, 615, 627, 630, 645, 651, 660, 663, 665, 690, 693, 705, 714, 715, 735, 741, 759, 765, 770, 777, 780

Offset

1,1

Comments

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).

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

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

Example

390 is a term since 390 = 2 * 3 * 5 * 13 has exactly 3 odd prime factors (3,5,13).
525 is a term since 525 = 3 * 5^2 * 7 has exactly 3 odd prime factors (3,5,7).

Maple

filter:= proc(n) local L;
  L:= ifactors(n)[2];
  nops(select(t -> t[1] <> 2, L)) = 3
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 01 2025

Mathematica

Select[Range[800], PrimeNu[# / 2^IntegerExponent[#, 2]] == 3 &] (* Amiram Eldar, Nov 21 2025 *)

Prog

(PARI) is(k) = (omega(k >> valuation(k, 2))==3); \\ Jason Yuen, Dec 24 2025

Crossrefs

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

Sequence in context: A203614 A252069 A133509 * A013590 A216918 A278569

Adjacent sequences: A386603 A386604 A386605 * A386607 A386608 A386609

Keyword

nonn

Author

A.H.M. Smeets, Nov 21 2025

Status

approved