A117203 -id:A117203 - cp's OEIS frontend

A117204 Squarefree positive integers k such that 2*k+1 is also squarefree.

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 23, 26, 29, 30, 33, 34, 35, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 59, 61, 65, 66, 69, 70, 71, 74, 77, 78, 79, 82, 83, 86, 89, 91, 93, 95, 97, 101, 102, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119

Offset: 1

Leroy Quet, Mar 02 2006

nonn

The asymptotic density of this sequence is (3/2)*A065474 = 0.4839511484... (Erdős and Ivić, 1987). - Amiram Eldar, Mar 02 2021

			10 and 2*10 +1 = 21 are both squarefree, so 10 is in the sequence.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, Vol. 51 (1987), pp. 45-91.

Cf. A005117, A117203, A117206.

Cf. A005384, A111153, A065474.

sfQ[n_]:=SquareFreeQ[n]&&SquareFreeQ[2n+1]; Select [Range[200],sfQ] (* Harvey P. Dale, Mar 12 2011 *)

a(n) = (A117203(n) - 1)/2.

More terms from Jonathan Vos Post, Mar 03 2006

Corrected and extended by Harvey P. Dale, Mar 12 2011

A117205 Odd squarefree positive integers k such that (k+1)/2 is also squarefree.

Original entry on oeis.org

1, 3, 5, 11, 13, 19, 21, 29, 33, 37, 41, 43, 51, 57, 59, 61, 65, 67, 69, 73, 77, 83, 85, 91, 93, 101, 105, 109, 113, 115, 123, 129, 131, 133, 137, 139, 141, 145, 155, 157, 163, 165, 173, 177, 181, 185, 187, 193, 201, 203, 205, 209, 211, 213, 217, 219, 221, 227, 229

Offset: 1

Leroy Quet, Mar 02 2006

nonn

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

			21 and (21+1)/2 = 11 are both squarefree, so 21 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős and Aleksandar Ivić, The distribution of values of a certain class of arithmetic functions at consecutive integers, Colloq. Math. Soc. János Bolyai, Vol. 51 (1987), pp. 45-91.

Cf. A005117, A065474, A117203, A117206.

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=1..300); # Emeric Deutsch, Mar 07 2006

Select[Range[1,301,2],And@@(SquareFreeQ/@{#,(#+1)/2})&] (* Harvey P. Dale, Jul 30 2013 *)

a(n) = 2 * A117206(n) - 1.

More terms from Emeric Deutsch, Mar 07 2006

cp's OEIS Frontend

A117204 Squarefree positive integers k such that 2*k+1 is also squarefree.

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