A117203 - OEIS

%I #26 Feb 17 2021 04:06:27

%S 3,5,7,11,13,15,21,23,29,31,35,39,43,47,53,59,61,67,69,71,77,79,83,85,

%T 87,93,95,103,107,111,115,119,123,131,133,139,141,143,149,155,157,159,

%U 165,167,173,179,183,187,191,195,203,205,211,213,215,219,221,223,227

%N Odd squarefree positive integers k such that (k-1)/2 is also squarefree.

%C The asymptotic density of this sequence is (3/4)*A065474 = 0.2419755742... (Erdős and Ivić, 1987). - _Amiram Eldar_, Feb 17 2021

%H Amiram Eldar, <a href="/A117203/b117203.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Erdős and Aleksandar Ivić, <a href="https://users.renyi.hu/~p_erdos/1987-32b.pdf">The distribution of values of a certain class of arithmetic functions at consecutive integers</a>, Colloq. Math. Soc. János Bolyai, Vol. 51 (1987), pp. 45-91.

%F a(n) = 2*A117204(n) + 1.

%e 21 and (21-1)/2 = 10 are both squarefree, so 21 is in the sequence.

%p with(numtheory): a:=proc(n) if n mod 2 =1 and abs(mobius(n))>0 and abs(mobius((n-1)/2))>0 then n else fi end: seq(a(n),n=2..300); # _Emeric Deutsch_, Mar 07 2006

%t fQ[n_] := Max @@ Last /@ FactorInteger@n < 2; Select[ 2Range@113 + 1, fQ@# && fQ[(# - 1)/2] &] (* _Robert G. Wilson v_, Apr 18 2006 *)

%o (PARI) for (i=1,100,if(issquarefree(2*i+1) && issquarefree(i),print1(2*i+1,","))) \\ Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 23 2006

%Y Cf. A005117, A065474, A117204, A117205.

%K nonn

%O 1,1

%A _Leroy Quet_, Mar 02 2006

%E More terms from _Emeric Deutsch_ and _Reinhard Zumkeller_, Mar 07 2006

%E More terms from Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 23 2006

%E More terms from _Robert G. Wilson v_, Apr 18 2006