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A364265
The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).
8
323567034, 431684330, 468780388, 481098980, 577922904, 639336984, 715008644, 720990620, 726167154, 735965384, 769385252, 808810638, 822981560, 831034918, 839075510, 847765554, 879549670, 895723268, 902976710, 903293468, 904796814, 918520420, 940737005, 944087484, 982059364
OFFSET
1,1
COMMENTS
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
LINKS
FORMULA
a(1) = A138206(3).
{k: A001221(k) = A001221(k+1) = A001221(k+2) = 6}.
MAPLE
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:
PROG
(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
CROSSREFS
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: A331446 A184573 A112429 * A104923 A153753 A035794
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jul 16 2023
EXTENSIONS
More terms from David A. Corneth, Jul 18 2023
STATUS
approved