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

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

Author

Juri-Stepan Gerasimov, May 06 2016

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A005117, A065474, A069977, A226993.

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

Python

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

Formula

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