A373666 Smallest positive integer whose square can be written as the sum of n positive perfect squares whose square roots differ by no more than 1.

1, 5, 3, 2, 5, 3, 4, 10, 3, 4, 7, 6, 4, 9, 6, 4, 25, 6, 5, 10, 6, 5, 16, 6, 5, 12, 6, 7, 11, 6, 7, 20, 6, 7, 15, 6, 7, 31, 9, 7, 13, 9, 7, 14, 9, 7, 36, 9, 7, 15, 9, 8, 22, 9, 8, 17, 9, 8, 16, 9, 8, 49, 9, 8, 20, 9, 10, 50, 9, 10, 17, 9, 10, 19, 9, 10, 28, 9

Offset

1,2

Comments

Shortest possible integer length of the diagonal of an n-dimensional hyperrectangle where each edge has a positive integer length, and edge lengths differ by no more than 1.

Links

Table of n, a(n) for n=1..78.

Formula

a(n) = min(d) such that d^2 - n*b^2 == 0 (mod 2*b + 1) and d >= ceiling(sqrt(n)) where b = floor(sqrt(d^2/n)).

Example

a(1) = 1 because 1^2 = 1^2.
a(2) = 5 because 5^2 = 3^2 + 4^2.
a(3) = 3 because 3^2 = 1^2 + 2*(2^2).
a(4) = 2 because 2^2 = 4*(1^2).
a(5) = 5 because 5^2 = 4*(2^2) + 3^2.
a(6) = 3 because 3^2 = 5*(1^2) + 2^2.
a(7) = 4 because 4^2 = 4*(1^2) + 3*(2^2).

Prog

(PARI) a(n) = d=ceil(sqrt(n)); while(true, b=floor(sqrt((d^2)/n)); if (((d^2)-(b^2)*n)%(b*2+1)==0, return(d), d++)) \\ Charles L. Hohn, Jul 02 2024

Crossrefs

Cf. A351061.

Sequence in context: A165102 A294568 A290349 * A009661 A023576 A271523

Adjacent sequences: A373663 A373664 A373665 * A373667 A373668 A373669

Keyword

nonn,changed

Author

Charles L. Hohn, Jun 12 2024

Status

approved