A383672 Squarefree numbers k such that k^2+1 is not squarefree.

7, 38, 41, 43, 57, 70, 82, 93, 107, 118, 143, 157, 182, 193, 218, 239, 251, 257, 282, 293, 307, 318, 327, 357, 382, 393, 407, 418, 437, 443, 457, 482, 493, 515, 518, 543, 557, 577, 582, 593, 606, 607, 618, 643, 682, 707, 718, 743, 746, 757, 782, 793, 807, 818, 829, 843, 857, 893

Offset

1,1

Links

Table of n, a(n) for n=1..58.

Example

38 = 2*19 is squarefree but 38*38 + 1 = 1445 = 5*17*17 is not squarefree.

Maple

filter:= proc(n) numtheory:-issqrfree(n) and not numtheory:-issqrfree(n^2+1) end proc:
select(filter, [$1..1000]); # Robert Israel, May 04 2025

Mathematica

Select[Range[900], SquareFreeQ[#] && !SquareFreeQ[#^2+1] &] (* Stefano Spezia, May 04 2025 *)

Prog

(Python)
from sympy import factorint
def is_squarefree(n):
    return all(exponent == 1 for exponent in factorint(n).values())
print([a for a in range(1, 900) if is_squarefree(a) and not(is_squarefree(a*a + 1))])
(PARI) isok(k) = issquarefree(k) && !issquarefree(k^2+1); \\ Michel Marcus, May 04 2025

Crossrefs

Intersection of A005117 and A049532.

Includes A141932 and A141941.

Cf. A080666, A224718.

Sequence in context: A127729 A129736 A220852 * A290176 A003352 A346278

Adjacent sequences: A383669 A383670 A383671 * A383673 A383674 A383675

Keyword

nonn

Author

Alexandre Herrera, May 04 2025

Status

approved