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
Keywords
Examples
38 = 2*19 is squarefree but 38*38 + 1 = 1445 = 5*17*17 is not squarefree.
Programs
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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
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Mathematica
Select[Range[900],SquareFreeQ[#] && !SquareFreeQ[#^2+1] &] (* Stefano Spezia, May 04 2025 *)
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PARI
isok(k) = issquarefree(k) && !issquarefree(k^2+1); \\ Michel Marcus, May 04 2025
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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))])