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

Author

Amiram Eldar, Dec 09 2024

Keywords

• nonn
• easy

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... .

Examples

			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.
		

Links

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

Crossrefs

Subsequence of A024619, A104210 and A378885.

Subsequences: A006094, A256617.

Cf. A008588, A020639, A151800.

Programs

Mathematica

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

PARI

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