A272799 - OEIS

%I #28 Apr 24 2024 17:02:41

%S 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,

%T 39,42,43,44,45,46,47,48,51,52,53,54,55,56,57,64,65,66,69,70,71,72,75,

%U 78,79,80,81,82,83,89,90,91,92,93,96,97,98,99,100,101,102,105,106,107,108,109,110

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

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

%H Robert Israel, <a href="/A272799/b272799.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = (A069977(n)+1)/2. - _Charles R Greathouse IV_, May 15 2016

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

%p Res:=  NULL: count:= 0: state:= 1;

%p for n from 1 while count < 100 do

%p   if numtheory:-issqrfree(2*n+1) then

%p     if state = 1 then Res:= Res, n; count:= count+1;

%p     else

%p       state:= 1;

%p     fi

%p   else

%p     state:= 0;

%p   fi

%p od:

%p Res; # _Robert Israel_, Apr 15 2019

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

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

%o (PARI) is(n)=issquarefree(2*n-1) && issquarefree(2*n+1) \\ _Charles R Greathouse IV_, May 15 2016

%o (Python)

%o from itertools import count, islice

%o from sympy import factorint

%o def A272799_gen(startvalue=1): # generator of terms >= startvalue

%o     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)))

%o A272799_list = list(islice(A272799_gen(),20)) # _Chai Wah Wu_, Apr 24 2024

%Y Cf. A005117, A065474, A069977, A226993.

%K nonn,easy

%O 1,2

%A _Juri-Stepan Gerasimov_, May 06 2016