A364265 - OEIS

%I #34 Jun 10 2024 00:07:01

%S 323567034,431684330,468780388,481098980,577922904,639336984,

%T 715008644,720990620,726167154,735965384,769385252,808810638,

%U 822981560,831034918,839075510,847765554,879549670,895723268,902976710,903293468,904796814,918520420,940737005,944087484,982059364

%N The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

%C 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

%H David A. Corneth, <a href="/A364265/b364265.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = A138206(3).

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

%p omega := proc(n)

%p     nops(numtheory[factorset](n)) ;

%p end proc:

%p for k from 1 do

%p     if omega(k) = 6 then

%p         if omega(k+1) = 6 then

%p             if omega(k+2) = 6 then

%p                 print(k) ;

%p             end if;

%p         end if;

%p     end if;

%p end do:

%o (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

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

%Y Cf. A364266 (5 distinct factors).

%Y See also A001221, A001222, A005117.

%Y 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).

%K nonn

%O 1,1

%A _R. J. Mathar_, Jul 16 2023

%E More terms from _David A. Corneth_, Jul 18 2023