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

Author

Benoit Cloitre, Jun 09 2002

Keywords

• nonn

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.

Crossrefs

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

Programs

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 *)

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

Formula

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