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

30, 60, 90, 105, 120, 150, 180, 210, 240, 270, 300, 315, 330, 360, 385, 390, 420, 450, 480, 510, 525, 540, 570, 600, 630, 660, 690, 720, 735, 750, 780, 810, 840, 870, 900, 930, 945, 960, 990, 1001, 1020, 1050, 1080, 1110, 1140, 1155, 1170, 1200, 1230, 1260, 1290

Offset: 1

Author

Amiram Eldar, Dec 09 2024

Keywords

• nonn
• easy

Comments

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

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

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... .

Examples

			60 = 2^2 * 3 * 5 is a term since 2, 3 and 5 are consecutive primes.
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.
1365 = 3 * 5 * 7 * 13 is a term since 3, 5 and 7 are consecutive primes.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A000977.

Subsequences: A046301, A378884.

Cf. A020639, A101300, A151800, A249674.

Programs

Mathematica

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 2 && p[[2]] == NextPrime[p[[1]]] && p[[3]] == NextPrime[p[[2]]]]; Select[Range[1300], q]

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));