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A378885 Numbers that are divisible by at least three different primes and the smallest three of them are consecutive primes.

%I A378885 #8 Dec 10 2024 09:59:02
%S A378885 30,60,90,105,120,150,180,210,240,270,300,315,330,360,385,390,420,450,
%T A378885 480,510,525,540,570,600,630,660,690,720,735,750,780,810,840,870,900,
%U A378885 930,945,960,990,1001,1020,1050,1080,1110,1140,1155,1170,1200,1230,1260,1290
%N A378885 Numbers that are divisible by at least three different primes and the smallest three of them are consecutive primes.
%C A378885 All the positive multiples of 30 (A249674 \ {0}) are terms.
%C A378885 Numbers k such that A151800(A020639(k)) | k and also A101300(A020639(k)) | k.
%C A378885 The asymptotic density of this sequence is Sum_{k>=1} (Product_{j=1..k-1} (1-1/prime(j))) / (prime(k)*prime(k+1)*prime(k+2)) = 0.03943839735407432193784... .
%H A378885 Amiram Eldar, <a href="/A378885/b378885.txt">Table of n, a(n) for n = 1..10000</a>
%e A378885 60 = 2^2 * 3 * 5 is a term since 2, 3 and 5 are consecutive primes.
%e A378885 770 = 2 * 5 * 7 * 11 is not a term since its smallest prime divisor is 2 and it is not divisible by 3, the prime next to 2.
%e A378885 1365 = 3 * 5 * 7 * 13 is a term since 3, 5 and 7 are consecutive primes.
%t A378885 q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 2 && p[[2]] == NextPrime[p[[1]]] && p[[3]] == NextPrime[p[[2]]]]; Select[Range[1300], q]
%o A378885 (PARI) is(k) = if(k == 1, 0, my(p = factor(k)[,1]); #p > 2 && p[2] == nextprime(p[1]+1) && p[3] == nextprime(p[2]+1));
%Y A378885 Subsequence of A000977.
%Y A378885 Subsequences: A046301, A378884.
%Y A378885 Cf. A020639, A101300, A151800, A249674.
%K A378885 nonn,easy
%O A378885 1,1
%A A378885 _Amiram Eldar_, Dec 09 2024