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A071870
Numbers k such that gpf(k) > gpf(k+1) > gpf(k+2) where gpf(k) denotes the largest prime factor of k.
12
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
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
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Jun 09 2002
STATUS
approved