A378885 - OEIS

%I #8 Dec 10 2024 09:59:02

%S 30,60,90,105,120,150,180,210,240,270,300,315,330,360,385,390,420,450,

%T 480,510,525,540,570,600,630,660,690,720,735,750,780,810,840,870,900,

%U 930,945,960,990,1001,1020,1050,1080,1110,1140,1155,1170,1200,1230,1260,1290

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

%C All the positive multiples of 30 (A249674 \ {0}) are terms.

%C Numbers k such that A151800(A020639(k)) | k and also A101300(A020639(k)) | k.

%C 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 Amiram Eldar, <a href="/A378885/b378885.txt">Table of n, a(n) for n = 1..10000</a>

%e 60 = 2^2 * 3 * 5 is a term since 2, 3 and 5 are consecutive primes.

%e 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 1365 = 3 * 5 * 7 * 13 is a term since 3, 5 and 7 are consecutive primes.

%t 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 (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 Subsequence of A000977.

%Y Subsequences: A046301, A378884.

%Y Cf. A020639, A101300, A151800, A249674.

%K nonn,easy,new

%O 1,1

%A _Amiram Eldar_, Dec 09 2024