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

1,2

Comments

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

Links

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

Formula

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

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) = 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).

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 *)

Crossrefs

Cf. A215539.

Sequence in context: A372389 A040022 A165100 * A159935 A165102 A294568

Adjacent sequences: A351058 A351059 A351060 * A351062 A351063 A351064

Keyword

nonn

Author

Lee A. Newberg, Jan 30 2022

Extensions

More terms from Jinyuan Wang, Jan 30 2022

Status

approved