A226993 -id:A226993 - cp's OEIS frontend

A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

2, 4, 6, 12, 14, 16, 18, 20, 22, 30, 32, 34, 36, 38, 40, 42, 52, 54, 56, 58, 60, 66, 68, 70, 72, 78, 84, 86, 88, 90, 92, 94, 96, 102, 104, 106, 108, 110, 112, 114, 128, 130, 132, 138, 140, 142, 144, 150, 156, 158, 160, 162, 164, 166, 178, 180, 182, 184, 186, 192, 194, 196, 198

Offset: 1

Lekraj Beedassy, Feb 25 2002

D is unique iff x^2 - 1 is squarefree, in which case it follows with necessity that D=x^2-1 and y=1.
All terms are even. A014574 is a subsequence.
Conjecture: All terms of A002110 > 1 are a subsequence. - Griffin N. Macris, Apr 11 2016
All n such that n+1 and n-1 are in A056911. - Robert Israel, Apr 12 2016
The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Feb 25 2021

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

Cf. A002110, A005117, A014574, A056911, A065474, A226993, A272799, A280892, A379971 (characteristic function).

Subsequence of A379965.

[n: n in [1..110] | IsSquarefree(n-1) and IsSquarefree(n+1)]; // Juri-Stepan Gerasimov, Jan 17 2017

select(t -> numtheory:-issqrfree(t^2-1), [seq(n,n=2..1000,2)]); # Robert Israel, Apr 12 2016

Select[Range[200], SquareFreeQ[#^2-1]&] (* Vladimir Joseph Stephan Orlovsky, Oct 26 2009 *)

from itertools import count, islice
from sympy import factorint
def A067874_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda k:max(factorint(k-1).values(),default=1)==1 and max(factorint(k+1).values())==1, count(max(startvalue+(startvalue&1),2),2))
A067874_list = list(islice(A067874_gen(),20)) # Chai Wah Wu, Apr 24 2024

a(n) = 2*A272799(n). - Juri-Stepan Gerasimov, Jan 17 2017

Corrected and extended by Max Alekseyev, Apr 26 2009

Further edited by Max Alekseyev, Apr 28 2009

A272799 Numbers k such that 2k - 1 and 2k + 1 are squarefree.

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 20, 21, 26, 27, 28, 29, 30, 33, 34, 35, 36, 39, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 64, 65, 66, 69, 70, 71, 72, 75, 78, 79, 80, 81, 82, 83, 89, 90, 91, 92, 93, 96, 97, 98, 99, 100, 101, 102, 105, 106, 107, 108, 109, 110

Offset: 1

Juri-Stepan Gerasimov, May 06 2016

The asymptotic density of this sequence is 2 * Product_{p prime} (1 - 2/p^2) = 2 * A065474 = 0.645268... . - Amiram Eldar, Feb 10 2021

			a(1) = 1 because 2*1 - 1 = 1 is squarefree and 2*1 + 1 = 3 is squarefree.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005117, A065474, A069977, A226993.

[n: n in [1..110] | IsSquarefree(2*n-1) and IsSquarefree(2*n+1)];

Res:=  NULL: count:= 0: state:= 1;
for n from 1 while count < 100 do
  if numtheory:-issqrfree(2*n+1) then
    if state = 1 then Res:= Res, n; count:= count+1;
    else
      state:= 1;
    fi
  else
    state:= 0;
  fi
od:
Res; # Robert Israel, Apr 15 2019

Select[Range[12^4], And[Or[# == 1, GCD @@ FactorInteger[#][[All, 2]] > 1], SquareFreeQ[# - 1], SquareFreeQ[# + 1]] &] (* Michael De Vlieger, May 08 2016 *)

is(n)=issquarefree(2*n-1) && issquarefree(2*n+1) \\ Charles R Greathouse IV, May 15 2016

from itertools import count, islice
from sympy import factorint
def A272799_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:max(factorint((k<<1)-1).values(),default=1)==1 and max(factorint((k<<1)+1).values())==1, count(max(startvalue,1)))
A272799_list = list(islice(A272799_gen(),20)) # Chai Wah Wu, Apr 24 2024

a(n) = (A069977(n)+1)/2. - Charles R Greathouse IV, May 15 2016

cp's OEIS Frontend

A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

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A272799 Numbers k such that 2k - 1 and 2k + 1 are squarefree.

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cp's OEIS Frontend

A067874 Positive integers x satisfying x^2 - D*y^2 = 1 for a unique integer D.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Formula

Extensions

A272799 Numbers k such that 2*k - 1 and 2*k + 1 are squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A272799 Numbers k such that 2k - 1 and 2k + 1 are squarefree.