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 1-3 of 3 results.

A088072 Numbers sandwiched between two numbers having only one prime divisor (at least) one of which is composite.

Original entry on oeis.org

3, 8, 10, 24, 26, 28, 48, 80, 82, 126, 168, 242, 360, 728, 840, 1368, 1848, 2208, 2400, 3720, 5040, 6562, 6858, 7920, 10608, 11448, 14640, 16128, 17160, 19320, 19682, 24390, 28560, 29790, 29928, 36480, 44520, 49728, 50652, 54288, 57120, 59050
Offset: 1

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Author

Amarnath Murthy, Sep 22 2003

Keywords

Comments

Terms of A088071 which are not terms of A014574 (average of twin prime pairs).
Conjecture (1): Sequence is infinite. Conjecture (2): There are only finitely many terms which are sandwiched between composite numbers (26 is the only such term below 1000000).

Examples

			8 is a term since 7 is prime and 9 = 3*3 is composite; 26 is a term since 25 = 5*5 and 27 = 3*3*3 are both composite.

Links

Crossrefs

Cf. A088071, A014574.

Programs

  • PARI
    {for(n=2,60000,if(matsize(factor(n-1))[1]*matsize(factor(n+1))[1]==1&&factor(n-1)[1,2]*factor(n+1)[1,2]>1,print1(n,",")))}

Extensions

Edited and extended by Klaus Brockhaus, Sep 25 2003

A120432 Numbers n such that n-1 and n+1 are prime powers.

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 12, 18, 24, 26, 28, 30, 42, 48, 60, 72, 80, 82, 102, 108, 126, 138, 150, 168, 180, 192, 198, 228, 240, 242, 270, 282, 312, 348, 360, 420, 432, 462, 522, 570, 600, 618, 642, 660, 728, 810, 822, 828, 840, 858, 882, 1020, 1032, 1050, 1062, 1092
Offset: 1

Views

Author

Greg Huber, Jul 13 2006

Keywords

Comments

A generalization of A014574.

Examples

			10 is in the sequence because both 9 and 11 are prime powers; 26 is in the sequence because both 25 and 27 are prime powers.

Links

Crossrefs

Cf. A000961, A014574.

Programs

  • Magma
    [2] cat [n : n in [3..1110] | IsPrimePower(n-1) and  IsPrimePower(n+1)]; // Vincenzo Librandi, Nov 25 2016
  • Maple
    with(numtheory): a:=proc(n) if nops(factorset(n-1))*nops(factorset(n+1))=1 then n else fi end: 2,seq(a(n),n=2..1500); # Emeric Deutsch, Jul 23 2006
  • Mathematica
    Insert[Select[Range[3, 3000], Length[FactorInteger[ # - 1]] == Length[ FactorInteger[ # + 1]] == 1 &], 2, 1] (* Stefan Steinerberger, Jul 23 2006 *)
Join[{2}, Select[Range[1100], And @@ PrimePowerQ /@ {# - 1, # + 1} &]] (* Ivan Neretin, Nov 24 2016 *)

Formula

{2} UNION A088071. - R. J. Mathar, Aug 07 2008

Extensions

More terms from Emeric Deutsch, Stefan Steinerberger and Ryan Propper, Jul 23 2006

A365416 Numbers k such that 2*k-1 and 2*k+1 are both prime powers (A246655).

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 12, 13, 14, 15, 21, 24, 30, 36, 40, 41, 51, 54, 63, 69, 75, 84, 90, 96, 99, 114, 120, 121, 135, 141, 156, 174, 180, 210, 216, 231, 261, 285, 300, 309, 321, 330, 364, 405, 411, 414, 420, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 684
Offset: 1

Views

Author

Jianing Song, Oct 22 2023

Keywords

Comments

According to Pillai's conjecture, k = 13 is the only term such that 2*k-1 and 2*k+1 both have exponent greater than 1.

Examples

			41 is a term since 2*41-1 = 81 is a prime power, and 2*41+1 = 83 is a prime.

Links

Crossrefs

Cf. A246655. Supersequence of A040040 and 2*A365411.
Cf. A088071, A175593.

Programs

  • PARI
    isA365416(n) = isprimepower(2*n-1) && isprimepower(2*n+1)
Showing 1-3 of 3 results.

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

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