A082418 -id:A082418 - cp's OEIS frontend

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

Benoit Cloitre, Jun 09 2002

nonn

Erdős conjectured that this sequence is infinite.

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

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

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.

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

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

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,",")))

gpf(n) = vecmax(factor(n)[,1]);
isok(k) = (gpf(k) > gpf(k+1)) && (gpf(k+1) > gpf(k+2)); \\ Michel Marcus, Nov 02 2020

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

A082417 Numbers k such that P(k) < P(k+1) > P(k+2), where P(k) is the largest prime factor of k (A006530).

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 18, 22, 25, 28, 30, 33, 36, 40, 42, 46, 48, 50, 52, 54, 58, 60, 64, 66, 68, 70, 72, 75, 78, 82, 85, 88, 93, 96, 98, 100, 102, 106, 108, 110, 112, 115, 117, 121, 126, 128, 130, 133, 136, 138, 141, 145, 148, 150, 154, 156, 160, 162, 166, 172, 178, 180, 182

Offset: 1

N. J. A. Sloane, Apr 25 2003

nonn

Antal Balog, On the largest prime factor of consecutive integers, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 337, #975-11-76.

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), p. 311-321. [alternate link]

Cf. A006530, A082418-A082422, A071869, A071870, A100392.

gpf[n_] := FactorInteger[n][[-1, 1]]; ind = Position[Differences[Array[gpf, 200]], ?(# > 0 &)] // Flatten; ind[[Position[Differences[ind], ?(# > 1 &)] // Flatten]] (* Amiram Eldar, Jun 06 2022 *)

a(n) = A100392(n) - 1. - T. D. Noe, Nov 26 2007

A100390 Numbers n where A006530 has a local minimum.

Original entry on oeis.org

4, 6, 8, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 42, 45, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 75, 77, 81, 84, 88, 90, 96, 98, 100, 102, 105, 108, 110, 112, 114, 117, 120, 125, 128, 130, 132, 135, 138, 140, 144, 147, 150, 154, 156, 160, 162, 165, 168, 175, 180

Offset: 1

Labos Elemer, Dec 14 2004

nonn

Powers of 2 and mostly largest powers of larger primes belong here.

			Let gpf(n) = A006530(n). 18 is in the sequence because gpf(17) = 17 > gpf(18) = 3 < gpf(19) = 19.

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

Cf. A006530.

mxp[x_] := Max[FactorInteger[x][[All, 1]]]; ta = {{0}}; Do[s1 = mxp[n - 1]; s = mxp[n]; s2 = mxp[n + 1]; If[Greater[s1, s] && Greater[s2, s], Print[{n, {s1, s, s2}}]; ta = Append[ta, n]], {n, 1, 512}]; ta = Delete[ta, 1] (* Original program edited by Michael De Vlieger, Mar 26 2017 *)
With[{n = 181}, Select[MapIndexed[{First@ #2 + 1, Differences@ #1} &, Partition[Array[FactorInteger[#][[-1, 1]] &, n], 3, 1]], Apply[And[#1 < 0, #2 > 0] &, Last@ #] &][[All, 1]]] (* Michael De Vlieger, Mar 26 2017 *)

a(n) = A082418(n) + 1. - T. D. Noe, Nov 26 2007

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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.

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A082417 Numbers k such that P(k) < P(k+1) > P(k+2), where P(k) is the largest prime factor of k (A006530).

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A100390 Numbers n where A006530 has a local minimum.

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