A280469 Numbers n such that n-1 and n+1 are squarefree and have the same number of prime factors.

4, 6, 12, 18, 30, 34, 42, 56, 60, 72, 86, 92, 94, 102, 108, 138, 142, 144, 150, 160, 180, 184, 186, 192, 198, 202, 204, 214, 216, 218, 220, 228, 236, 240, 248, 266, 270, 282, 300, 302, 304, 312, 320, 322, 328, 340, 348, 392, 394, 412, 414, 416, 420, 432, 446

Offset: 1

Author

Rick L. Shepherd, Jan 03 2017

Keywords

• nonn

Comments

For a given term n of this sequence, n-1 and n+1 are both squarefree k-almost primes for the same k. The sequence is thus the union of the averages (arithmetic means) of twin prime pairs (A014574), the averages of twin squarefree semiprime pairs, the averages of twin squarefree 3-almost prime pairs, ... (where "twin ... pairs" means the members of each pair differ by two). A subsequence of A280382 and of A280383.

Examples

			The number 34 is a term because 33 = 3*11 and 35 = 5*7, a twin semiprime pair. Unlike A280382 and A280383, 19 is not a term here because 18 = 2*3^2 and 20  = 2^2*5, neither of which is squarefree.
		

Links

• Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014574, A280382, A280383.

Programs

Mathematica

Select[Range@ 500, And[Times @@ First@ # == 1, SameQ @@ Last@ #] &@ Transpose@ Map[{Boole@ SquareFreeQ@ #, PrimeNu@ #} &, # + {-1, 1}] &] (* Michael De Vlieger, Jan 30 2017 *)

PARI

IsInA280469(n) = n > 1 && issquarefree(n-1) && issquarefree(n+1) && omega(n-1) == omega(n+1)