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

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

Author

Leroy Quet, Mar 02 2006

Keywords

• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

Cf. A005117, A065474, A117203, A117206.

Programs

Maple

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

Mathematica

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

Formula

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

Extensions

More terms from Emeric Deutsch, Mar 07 2006