A071869 Numbers k such that gpf(k) < gpf(k+1) < gpf(k+2) where gpf(k) denotes the largest prime factor of k.

8, 9, 20, 21, 24, 27, 32, 45, 56, 57, 77, 81, 84, 90, 91, 92, 105, 114, 120, 125, 132, 135, 140, 144, 147, 165, 168, 169, 170, 171, 175, 176, 177, 189, 200, 204, 212, 216, 220, 221, 225, 231, 234, 235, 247, 252, 260, 261, 275, 288, 289, 300, 315, 324, 345, 354

Offset

1,1

Comments

Erdős and Pomerance showed in 1978 that this sequence is infinite.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Paul Erdős and Carl Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), pp. 311-321.

Formula

a(n) = A079747(n+1) - 1. - T. D. Noe, Nov 26 2007

Mathematica

gpf[n_] := FactorInteger[n][[-1, 1]]; ind = Position[Differences[Array[gpf, 350, 2]], _?(# > 0 &)] // Flatten; ind[[Position[Differences[ind], 1] // Flatten]] + 1 (* Amiram Eldar, Jun 05 2022 *)

Prog

(PARI) for(n=2, 500, if(sign(component(component(factor(n), 1), omega(n))-component(component(factor(n+1), 1), omega(n+1)))+sign(component(component(factor(n+1), 1), omega(n+1))-component(component(factor(n+2), 1), omega(n+2)))==-2, print1(n, ", ")))
(Python)
from sympy import factorint
A071869_list, p, q, r = [], 1, 2, 3
for n in range(2, 10**4):
    p, q, r = q, r, max(factorint(n+2))
    if p < q < r:
        A071869_list.append(n) # Chai Wah Wu, Jul 24 2017

Crossrefs

Cf. A006530, A070089, A071870, A079747, A082417-A082422.

Sequence in context: A061414 A240915 A281225 * A326444 A309484 A308989

Adjacent sequences: A071866 A071867 A071868 * A071870 A071871 A071872

Keyword

nonn

Author

Benoit Cloitre, Jun 09 2002

Status

approved