A117203 Odd squarefree positive integers k such that (k-1)/2 is also squarefree.

3, 5, 7, 11, 13, 15, 21, 23, 29, 31, 35, 39, 43, 47, 53, 59, 61, 67, 69, 71, 77, 79, 83, 85, 87, 93, 95, 103, 107, 111, 115, 119, 123, 131, 133, 139, 141, 143, 149, 155, 157, 159, 165, 167, 173, 179, 183, 187, 191, 195, 203, 205, 211, 213, 215, 219, 221, 223, 227

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 = 10 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, A117204, A117205.

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=2..300); # Emeric Deutsch, Mar 07 2006

Mathematica

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

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

Formula

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

Extensions

More terms from Emeric Deutsch and Reinhard Zumkeller, Mar 07 2006

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

More terms from Robert G. Wilson v, Apr 18 2006