A391137 Numbers k such that the divisors of k^2 + 1 are all of the form m^2 + 1.

0, 1, 2, 3, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40, 54, 56, 66, 74, 84, 90, 94, 110, 116, 120, 124, 126, 130, 134, 146, 150, 156, 160, 170, 176, 180, 184, 204, 206, 210, 224, 230, 236, 240, 250, 256, 260, 264, 270, 280, 284, 300, 306, 314, 326, 340, 350, 384, 386, 396, 400

Offset

1,3

Comments

It appears to be the union of {0, 3} and A005574.

Links

Table of n, a(n) for n=1..60.

Example

3 is a term since the proper divisors of 3^2 + 1 = 10 are 1 = 0^2 + 1, 2 = 1^1 + 1, 5 = 2^2 + 1;
6 is a term since 6^2 + 1 = 37 is prime and the only proper divisor 1 = 0^2 + 1;
8 is not a term since the divisor 13 of 8^2 + 1 = 65 is not of the form m^2 + 1.

Mathematica

Sqrt[Select[Range[0, 400]^2, AllTrue[Sqrt[Divisors[#+1]-1], IntegerQ]&]]

Crossrefs

Cf. A002496, A002522, A005574, A027750, A027751, A193432.

Sequence in context: A121152 A229863 A215255 * A381223 A200928 A318558

Adjacent sequences: A391134 A391135 A391136 * A391138 A391139 A391140

Keyword

nonn,easy

Author

Stefano Spezia, Nov 30 2025

Status

approved