1,1

To distinguish this from A259349: "Numbers n with exactly k distinct prime factors" means numbers with A001221(n) = omega(n) = k, which specifies that in the prime factorization n = Product_{i>=1} p_i^(e_i), e_i >= 1, the exponents are ignored, and only the size of the set of the (distinct) p_i is considered. In A259349, the numbers n are products of k distinct primes, which means in the prime factorization of n, all exponents e_i are equal to 1. (If all exponents e_i = 1, the n are squarefree, i.e., in A005117.) Rephrased: the n which are products of k distinct primes have A001221(n) = omega(n) = A001222(n) = bigomega(n) = k, whereas the n which have exactly k distinct prime factors are the superset of (weaker) requirement A001221(n) = omega(n) = k. - R. J. Mathar, Jul 18 2023

David A. Corneth, Table of n, a(n) for n = 1..10000

a(1) = A138206(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 6}.

omega := proc(n)

nops(numtheory[factorset](n)) ;

end proc:

for k from 1 do

if omega(k) = 6 then

if omega(k+1) = 6 then

if omega(k+2) = 6 then

print(k) ;

end if;

end if;

end if;

end do:

(PARI) upto(n) = {my(res = List(), streak = 0); forfactored(i = 2, n, if(#i[2]~ == 6, streak++; if(streak >= 3, listput(res, i[1] - 2)), streak = 0)); res} \\ David A. Corneth, Jul 18 2023

Cf. A259349 (requires squarefree). Subsequence of A273879.

Cf. A364266 (5 distinct factors).

See also A001221, A001222, A005117.

Numbers divisible by d distinct primes: A246655 (d=1), A007774 (d=2), A033992 (d=3), A033993 (d=4), A051270 (d=5), A074969 (d=6), A176655 (d=7), A348072 (d=8), A348073 (d=9).

Sequence in context: A184573 A393550 A112429 * A104923 A153753 A035794

Adjacent sequences: A364262 A364263 A364264 * A364266 A364267 A364268

R. J. Mathar, Jul 16 2023

More terms from David A. Corneth, Jul 18 2023

approved