A374805 a(n) is the smallest positive integer whose square can be represented as the sum of 3 distinct nonzero squares in exactly n ways, or -1 if no such number exists.

1, 7, 15, 23, 31, 21, 33, 39, 49, 45, 79, 57, 95, 103, 75, 69, 127, 87, 63, 151, 93, 167, 111, 123, 99, 187, 117, 105, 161, 241, 141, 135, 153, 247, 271, 283, 177, 183, 165, 275, 147, 171, 323, 219, -1, 213, 319, 379, 383, 255, 237, 225, 207, 267, 431, 329

Offset

0,2

Comments

From Zhao Hui Du, Oct 10 2025 (Start)

Let 6r(m) be the number of integral solutions to x^2 + y^2 + z^2 = m^2,

2s(m) be the number of integral solutions to x^2 + 2y^2 = m^2,

4t(m) be the number of integral solutions to x^2 + y^2 = m^2.

Then the number of positive integral solutions with 0<x<y<z and x^2+y^2+z^2 = m^2 is f(m) = (6r(m)-12s(m)-12t(m)+18)/48.

Since f(2m)=f(m), a(n) must be an odd positive integer or -1.

The Olds paper gives formula of r(m) and for odd m, it is easy to prove r(m)>=m.

In BBS, we proved s(m) <= 2m^0.75 and t(m) <= 2m^0.75, so that we could give lower bound of f(m). (End)

Links

Zhao Hui Du, Table of n, a(n) for n = 0..9999

C. D. Olds, On the Number of Representations of the Square of an Integer as the Sum of Three Squares, American Journal of Mathematics, Vol. 63, No. 4 (Oct., 1941), pp. 763-767 (5 pages).

Zhining Yang, a^2+b^2+c^2=d^2 with exact 44 solutions (Chinese)

Index entries for sequences related to sums of squares

Example

a(3) = 23: 23^2 = 3^2 + 6^2 + 22^2 = 3^2 + 14^2 + 18^2 = 6^2 + 13^2 + 18^2.

Crossrefs

Cf. A006339, A025415, A025442, A346071.

Sequence in context: A004215 A389598 A206906 * A295669 A179890 A292639

Adjacent sequences: A374802 A374803 A374804 * A374806 A374807 A374808

Keyword

sign

Author

Ilya Gutkovskiy, Jul 20 2024

Extensions

a(35)-a(43) from Michael S. Branicky, Jul 21 2024

More terms from Zhao Hui Du, Oct 10 2025

Status

approved