A071869 -id:A071869 - 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

A082422 Numbers n such that P(n) > P(n+2) > P(n+1), where P(n) = largest prime factor of n (A006530).

Original entry on oeis.org

7, 19, 23, 26, 31, 47, 53, 67, 74, 76, 83, 89, 97, 109, 113, 119, 124, 127, 131, 134, 139, 146, 159, 167, 174, 181, 183, 188, 199, 207, 211, 215, 219, 233, 244, 246, 251, 259, 263, 274, 287, 293, 303, 307, 314, 323, 327, 337, 339, 349, 353, 359, 362, 367, 379, 383, 386

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, A082417-A082421, A071869, A071870.

With[{lfs=Table[FactorInteger[n][[-1,1]],{n,400}]},Flatten[ Position[ Partition[ lfs,3,1],?(#[[1]]>#[[3]]>#[[2]]&),{1},Heads->False]]] (* _Harvey P. Dale, Sep 25 2014 *)

A100384 a(n) = the smallest number x >= 2 such that for m = x to x + n - 1, A006530(m) increases.

Original entry on oeis.org

2, 2, 8, 8, 90, 168, 9352, 46189, 721970, 721970, 6449639, 565062156, 11336460025, 37151747513, 256994754033

Offset: 1

Labos Elemer, Dec 09 2004

A006530(m) is the largest prime factor of m.
a(16) > 3*10^11. - Donovan Johnson, Oct 24 2009
a(16) > 10^13. - Giovanni Resta, Jul 25 2013

			a(5)=90 because the largest prime factors of 90,91,92,93,94 are 5,13,23,31,47.

Cf. A006530, A070089, A071869, A100376, A100385, A358890.

from sympy import factorint
def A100384(n):
    k, a = 2, [max(factorint(m+2)) for m in range(n)]
    while True:
        for i in range(1, n):
            if a[i-1] >= a[i]:
                break
        else:
            return k
        a = a[i:] + [max(factorint(k+j+n)) for j in range(i)]
        k += i # Chai Wah Wu, Jul 24 2017

Edited by Don Reble, Jun 13 2007
a(13)-a(15) from Donovan Johnson, Oct 24 2009
Name clarified by Peter Munn, Dec 05 2022

A079747 Numbers k such that gpf(k-1) < gpf(k) < gpf(k+1), where gpf(k) is the greatest prime factor of k (A006530).

Original entry on oeis.org

2, 9, 10, 21, 22, 25, 28, 33, 46, 57, 58, 78, 82, 85, 91, 92, 93, 106, 115, 121, 126, 133, 136, 141, 145, 148, 166, 169, 170, 171, 172, 176, 177, 178, 190, 201, 205, 213, 217, 221, 222, 226, 232, 235, 236, 248, 253, 261, 262, 276, 289, 290, 301, 316, 325, 346

Offset: 1

Reinhard Zumkeller, Jan 10 2003

nonn

Numbers k such that A079748(k-1) > 1.

			k=25: 25-1 = 24 = 3*2^3, 25 = 5^2 and 25+1 = 26 = 13*2, therefore 25 is a term (3 < 5 < 13).

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

Cf. A071869, A071870, A082417-A082422.

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

a(n) = A071869(n-1) + 1. - T. D. Noe, Nov 26 2007

A082418 Numbers n such that P(n) > P(n+1) < P(n+2), where P(n) = largest prime factor of n (A006530).

Original entry on oeis.org

3, 5, 7, 11, 15, 17, 19, 23, 26, 29, 31, 35, 39, 41, 44, 47, 49, 51, 53, 55, 59, 63, 65, 67, 69, 71, 74, 76, 80, 83, 87, 89, 95, 97, 99, 101, 104, 107, 109, 111, 113, 116, 119, 124, 127, 129, 131, 134, 137, 139, 143, 146, 149, 153, 155, 159, 161, 164, 167, 174, 179, 181, 183

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, A082417-A082422, A071869, A071870.

Flatten[Position[Partition[Table[FactorInteger[n][[-1,1]],{n,200}],3,1], ?(#[[1]]> #[[2]] <#[[3]]&),{1},Heads->False]] (* _Harvey P. Dale, Jan 10 2016 *)

a(n) = A100390(n) - 1. - T. D. Noe, Nov 26 2007

A100376 a(n) is the largest number x such that for m=n to n+x-1, A006530(m) increases.

Original entry on oeis.org

2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 3, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 3, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 1, 1, 2, 1, 5, 4, 3, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 3

Offset: 2

Labos Elemer, Dec 09 2004

nonn

A006530(m) is the greatest prime factor (gpf) of m.

If p is an odd prime, a(p)=1, because the largest prime factor of p+1 is smaller than p.

			a(8)=4 because the largest prime factors of 8,9,10,11 are 2,3,5,11; but gpf(12)=3.
From _Michael De Vlieger_, Jul 30 2017: (Start)
Value  First position
   1         3
   2         2
   3         9
   4         8
   5        90
   6       168
   7      9352
   8     46189
   9    721971
  10    721970
(End)

Michel Marcus, Table of n, a(n) for n = 2..10000

Cf. A006530, A070089, A071869, A100384, A100387.

With[{s = Differences@ Array[FactorInteger[#][[-1, 1]] &, 115]}, Table[1 + LengthWhile[Drop[s, n], # > 0 &], {n, Length@ s - 10}]] (* Michael De Vlieger, Jul 30 2017 *)

a(n) = {m = n+1; gpf = vecmax(factor(n)[,1]); while((ngpf=vecmax(factor(m)[,1])) > gpf, m++; gpf = ngpf;); m - n;} \\ Michel Marcus, Jul 25 2017

Edited by Don Reble, Jun 13 2007

A082421 Numbers n such that P(n+1) < P(n) < P(n+2), where P(n) = largest prime factor of n (A006530).

Original entry on oeis.org

3, 5, 11, 15, 17, 29, 35, 39, 41, 44, 49, 51, 55, 59, 63, 65, 69, 71, 80, 87, 95, 99, 101, 104, 107, 111, 116, 129, 137, 143, 149, 153, 155, 161, 164, 179, 186, 191, 195, 197, 203, 209, 224, 227, 230, 237, 239, 242, 249, 255, 265, 267, 269, 272, 279, 281, 285, 291, 296, 299

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, A082417-A082422, A071869, A071870.

Flatten[Position[Partition[Table[FactorInteger[n][[-1,1]],{n,400}],3,1],?(#[[2]]<#[[1]]<#[[3]]&),{1},Heads->False]] (* _Harvey P. Dale, Sep 27 2013 *)

A082419 Numbers n such that P(n+2) < P(n) < P(n+1), where P(n) = largest prime factor of n (A006530).

Original entry on oeis.org

6, 10, 22, 25, 28, 30, 33, 46, 52, 58, 68, 70, 78, 82, 88, 93, 98, 102, 106, 110, 115, 126, 130, 133, 138, 141, 145, 148, 160, 166, 172, 178, 187, 190, 198, 201, 205, 208, 213, 222, 226, 232, 236, 238, 248, 253, 262, 268, 273, 286, 292, 304, 306, 310, 316, 328, 346, 348

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, A082417-A082422, A071869, A071870.

pflist=Select[Partition[Table[{n,lpf[n]},{n,400}],3,1],#[[3,-1]]< #[[1,-1]]< #[[2,-1]]&]; Table[pflist[[n,1,1]],{n,Length[pflist]}] (* Harvey P. Dale, Oct 25 2011 *)

A082420 Numbers n such that P(n) < P(n+2) < P(n+1), where P(n) = largest prime factor of n (A006530).

Original entry on oeis.org

4, 12, 16, 18, 36, 40, 42, 48, 50, 54, 60, 64, 66, 72, 75, 85, 96, 100, 108, 112, 117, 121, 128, 136, 150, 154, 156, 162, 180, 182, 184, 192, 196, 210, 217, 228, 240, 243, 245, 250, 256, 264, 266, 270, 276, 280, 282, 290, 294, 297, 301, 308, 312, 320, 325, 330, 333, 336

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, A082417-A082422, A071869, A071870.

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.

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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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A082422 Numbers n such that P(n) > P(n+2) > P(n+1), where P(n) = largest prime factor of n (A006530).

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A100384 a(n) = the smallest number x >= 2 such that for m = x to x + n - 1, A006530(m) increases.

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A079747 Numbers k such that gpf(k-1) < gpf(k) < gpf(k+1), where gpf(k) is the greatest prime factor of k (A006530).

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A082418 Numbers n such that P(n) > P(n+1) < P(n+2), where P(n) = largest prime factor of n (A006530).

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A100376 a(n) is the largest number x such that for m=n to n+x-1, A006530(m) increases.

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A082421 Numbers n such that P(n+1) < P(n) < P(n+2), where P(n) = largest prime factor of n (A006530).

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