A378884 Numbers that are not powers of primes and whose two smallest prime divisors are consecutive primes.

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 143, 144, 150, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 234, 240, 245, 246, 252, 255, 258

Offset

1,1

Comments

Subsequence of A104210 and first differs from at an n = 15: A104210(15) = 70 = 2 * 5 * 7 is not a term of this sequence.

All the positive multiples of 6 (A008588 \ {0}) are terms.

Numbers k such that nextprime(lpf(k)) = A151800(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)) = 0.2178590011934... .

Links

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

Example

12 = 2^2 * 3 is a term since 2 and 3 are consecutive primes.
70 = 2 * 5 * 7 is not a term since 2 and 5 are not consecutive primes.
165 = 3 * 5 * 11 is a term since 3 and 5 are consecutive primes.

Mathematica

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 1 && p[[2]] == NextPrime[p[[1]]]]; Select[Range[300], q]

Prog

(PARI) is(k) = if(k == 1, 0, my(p = factor(k)[, 1]); #p > 1 && p[2] == nextprime(p[1]+1));

Crossrefs

Subsequence of A024619, A104210 and A378885.

Subsequences: A006094, A256617.

Cf. A008588, A020639, A151800.

Sequence in context: A091011 A324771 A104210 * A356736 A066312 A309944

Adjacent sequences: A378881 A378882 A378883 * A378885 A378886 A378887

Keyword

nonn,easy,new

Author

Amiram Eldar, Dec 09 2024

Status

approved