A351061 Smallest positive integer whose square can be written as the sum of n positive perfect squares.

1, 5, 3, 2, 4, 3, 4, 4, 3, 4, 5, 6, 4, 5, 6, 4, 5, 6, 5, 6, 6, 5, 7, 6, 5, 7, 6, 6, 7, 6, 7, 7, 6, 7, 7, 6, 7, 7, 8, 7, 7, 8, 7, 8, 8, 7, 8, 8, 7, 8, 9, 8, 8, 9, 8, 8, 9, 8, 9, 9, 8, 9, 9, 8, 9, 9, 9, 10, 9, 9, 10, 9, 9, 10, 9, 10, 10, 9, 10, 10, 9, 10, 10, 10

Offset: 1

Author

Lee A. Newberg, Jan 30 2022

Keywords

• nonn

Comments

Shortest possible integer length of the diagonal of an n-dimensional hyperrectangle where each edge has a positive integer length.

Examples

			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) = 4 because 4^2 = 3*(1^2) + 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).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A215539.

Programs

Maple

b:= proc(n, i, t) option remember; n>=t and (n=t or
      (i>0 and (b(n, i-1, t) or i^2<=n and b(n-i^2, i, t-1))))
    end:
a:= proc(n) option remember; local k;
      for k while not b(k^2, k, n) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 31 2022

Mathematica

b[n_, i_, t_] := b[n, i, t] = n >= t && (n == t ||
     (i > 0 && (b[n, i - 1, t] || i^2 <= n && b[n - i^2, i, t - 1])));
a[n_] := a[n] = Module[{k}, For[k = 1, !b[k^2, k, n], k++]; k];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 14 2022, after Alois P. Heinz *)

Formula

a(n) = sqrt(A215539(n)). - Jinyuan Wang, Jan 30 2022

Extensions

More terms from Jinyuan Wang, Jan 30 2022