cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A071869 Numbers k such that gpf(k) < gpf(k+1) < gpf(k+2) where gpf(k) denotes the largest prime factor of k.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Jun 09 2002

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • 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,",")))
  • Python
    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

Formula

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

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

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Jun 09 2002

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

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

Views

Author

N. J. A. Sloane, Apr 25 2003

Keywords

References

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

Links

Crossrefs

Cf. A006530, A082417-A082421, A071869, A071870.

Programs

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

Views

Author

Reinhard Zumkeller, Jan 10 2003

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

Cf. A071869, A071870, A082417-A082422.

Programs

Formula

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

Views

Author

N. J. A. Sloane, Apr 25 2003

Keywords

References

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

Links

Crossrefs

Cf. A006530, A082417-A082422, A071869, A071870.

Programs

Formula

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

Views

Author

N. J. A. Sloane, Apr 25 2003

Keywords

References

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

Links

Crossrefs

Cf. A006530, A082417-A082422, A071869, A071870.

Programs

A100392 Numbers k such that A006530(k-1) < A006530(k) > A006530(k+1).

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 26, 29, 31, 34, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 76, 79, 83, 86, 89, 94, 97, 99, 101, 103, 107, 109, 111, 113, 116, 118, 122, 127, 129, 131, 134, 137, 139, 142, 146, 149, 151, 155, 157, 161, 163, 167, 173, 179
Offset: 1

Views

Author

Labos Elemer, Dec 14 2004

Keywords

Comments

A006530(k) is the largest prime factor of k.
The sequence contains all odd primes.

Examples

			26 is here because the largest prime factors of 25, 26, 27 are 5, 13, 3.

Links

Crossrefs

Cf. A006530, A100390, A100393.

Programs

  • Mathematica
    <?(First[#]<#[[2]]>Last[#]&),{1},Heads-> False]]+ 1] (* _Harvey P. Dale, Feb 22 2014 *)

Formula

a(n) = A082417(n) + 1. - T. D. Noe, Nov 26 2007

Extensions

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

Views

Author

N. J. A. Sloane, Apr 25 2003

Keywords

References

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

Links

Crossrefs

Cf. A006530, A082417-A082422, A071869, A071870.

Programs

  • Mathematica
    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

Views

Author

N. J. A. Sloane, Apr 25 2003

Keywords

References

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

Links

Crossrefs

Cf. A006530, A082417-A082422, A071869, A071870.
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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