

A272799


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


3



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
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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*n1) and IsSquarefree(2*n+1)];
(PARI) is(n)=issquarefree(2*n1) && issquarefree(2*n+1) \\ Charles R Greathouse IV, May 15 2016


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

JuriStepan Gerasimov, May 06 2016


STATUS

approved



