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

1,2

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

Links

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A005117, A065474, A069977, A226993.

Sequence in context: A050023 A047505 A039071 * A300157 A265334 A050019

Adjacent sequences: A272796 A272797 A272798 * A272800 A272801 A272802

Keyword

nonn,easy

Author

Juri-Stepan Gerasimov, May 06 2016

Status

approved