A071870 Numbers k such that gpf(k) > gpf(k+1) > gpf(k+2) where gpf(k) denotes the largest prime factor of k.

13, 14, 34, 37, 38, 43, 61, 62, 73, 79, 86, 94, 103, 118, 122, 123, 142, 151, 152, 157, 158, 163, 173, 185, 193, 194, 202, 206, 214, 218, 223, 229, 241, 254, 257, 258, 271, 277, 278, 283, 284, 295, 298, 302, 313, 317, 318, 321, 322, 326, 331, 334, 341, 373

Offset

1,1

Comments

Erdős conjectured that this sequence is infinite.

Balog (2001) proved that this sequence is infinite. - Amiram Eldar, Aug 02 2020

Links

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

Antal Balog, On triplets with descending largest prime factors, Studia Scientiarum Mathematicarum Hungarica, Vol. 38, No. 1-4 (2001), pp. 45-50.

P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), pp. 311-321. [alternate link]

Sungjin Kim, Two Remarks on the Largest Prime Factors of n and n+1, J. of Integer Sequences, 23 (2020), #20.10.1.

Example

13 is a term since gpf(13) = 13, gpf(14) = 7, gpf(15) = 5, and 13 > 7 > 5.

Mathematica

Select[ Range[400], FactorInteger[#][[-1, 1]] >  FactorInteger[# + 1][[-1, 1]] > FactorInteger[# + 2][[-1, 1]] &] (* Jean-François Alcover, Jun 17 2013 *)

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, ", ")))
(PARI) gpf(n) = vecmax(factor(n)[, 1]);
isok(k) = (gpf(k) > gpf(k+1)) && (gpf(k+1) > gpf(k+2)); \\ Michel Marcus, Nov 02 2020
(Python)
from sympy import factorint
A071870_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:
        A071870_list.append(n) # Chai Wah Wu, Jul 24 2017

Crossrefs

Cf. A006530, A070087, A071869, A082417, A082418, A082419, A082420, A082421, A082422.

Sequence in context: A167996 A308122 A292116 * A346544 A041350 A041348

Adjacent sequences: A071867 A071868 A071869 * A071871 A071872 A071873

Keyword

nonn

Author

Benoit Cloitre, Jun 09 2002

Status

approved