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

1,1

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... .

Links

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

Example

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.

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]

Prog

(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));

Crossrefs

Subsequence of A000977.

Subsequences: A046301, A378884.

Cf. A020639, A101300, A151800, A249674.

Sequence in context: A222618 A325992 A056954 * A377952 A246947 A226944

Adjacent sequences: A378882 A378883 A378884 * A378886 A378887 A378888

Keyword

nonn,easy,new

Author

Amiram Eldar, Dec 09 2024

Status

approved