A071870 -id:A071870 - cp's OEIS frontend

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

Benoit Cloitre, Jun 09 2002

nonn

Erdős and Pomerance showed in 1978 that this sequence is infinite.

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.

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

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 *)

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,",")))

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

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

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 *)

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

A100385 a(n) is the least number x >= 2 such that for m=x to x+n-1, A006530(m) decreases.

Original entry on oeis.org

2, 3, 13, 13, 491, 1851, 12721, 12721, 109453, 586951, 120797465, 624141002, 4044619541, 267793490438, 315400191511, 1285600699441

Offset: 1

Labos Elemer, Dec 09 2004

A006530(m) is the largest prime factor of m.
a(15) > 3*10^11. - Donovan Johnson, Oct 24 2009
a(17) > 7*10^12. - Giovanni Resta, May 04 2017

			a(5)=491 because the largest prime factors of 491,492,493,494,495 are 491,41,29,19,11.

Cf. A006530, A070087, A070089, A071870, A100384, A100386.

Function[s, Prepend[Reverse@ FoldList[If[#2 > #1, #1, #2] &, Reverse@ #], 2] &@ Map[Function[k, First@ SelectFirst[s, And[Sign@ First@ # == 1, Length@ # == k] &]], Range[Max@ Map[Length, s]]]]@ SplitBy[Flatten[ Partition[Array[{#, FactorInteger[#][[-1, 1]]} &, 10^6], 2, 1] /. {{n_, a_}, {, b}} /; n > 0 :> -n Sign[Differences@ {a, b}]], Sign] (* Michael De Vlieger, May 04 2017, Version 10.2 *)

a(n) = A070089(x)+1, where x is the smallest positive integer such that A070089(x+1)-A070089(x) >= n. - Pontus von Brömssen, Nov 09 2022

Edited by Don Reble, Jun 13 2007
a(14) from Donovan Johnson, Oct 24 2009
a(15)-a(16) from Giovanni Resta, May 04 2017

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 *)

A100386 Numbers n such that for m=n to n+9, A006530(m) decreases.

Original entry on oeis.org

586951, 1473257, 4982941, 13565441, 24954141, 25384714, 26576686, 32026196, 35797623, 35953989, 37972276, 39048260, 51755761, 58769257, 60682681, 71342703, 77863117, 80826231, 84766857, 89768134, 98363506, 110482826, 115045547, 115898807, 120797465

Offset: 1

Labos Elemer, Dec 09 2004

nonn

A006530(n) is the largest prime factor of n.

			586951 is here because the largest prime factors of 586951..586960 are 586951,73369,21739,9467,1319,1193,1181,1091,677,29.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A006530, A070087, A071870, A100385.

<?(Max[Differences[#]]<0&),{1},Heads->False]//Flatten (* _Harvey P. Dale, Sep 18 2016 *)

Edited by Don Reble, Jun 13 2007

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

A071869 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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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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A100385 a(n) is the least number x >= 2 such that for m=x to x+n-1, A006530(m) decreases.

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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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A100386 Numbers n such that for m=n to n+9, A006530(m) decreases.

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