A161882 Smallest k such that n^2 = a_1^2 + ... + a_k^2 and all a_i are positive integers less than n.

4, 3, 4, 2, 3, 3, 4, 3, 2, 3, 3, 2, 3, 2, 4, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 3, 4, 2, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2

Offset

2,1

Comments

Related to hypotenuse numbers: A161882(A009003(n))=2 for all n.

Jacobi's four-square theorem can be used to show that a(n) <= 4. - Charles R Greathouse IV, Jul 31 2011

Links

Alois P. Heinz, Table of n, a(n) for n = 2..700

Jean-Charles Meyrignac, Computing minimal equal sums of like powers.

Eric Weisstein's World of Mathematics, Diophantine Equation 2nd Powers.

Formula

a(n)=2 iff n is in A009003 (hypotenuse numbers), a(n)=4 iff n is in A000079 (powers of 2), otherwise a(n)=3. - M. F. Hasler, Dec 17 2014

Example

2^2 = 1^2 + 1^2 + 1^2 + 1^2, so a(2)=4.
3^2 = 2^2 + 2^2 + 1^2, so a(3)=3.

Mathematica

f[n_, k_] := Select[PowersRepresentations[n^2, k, 2], AllTrue[#, 0<#<n&]&];
a[n_] := For[k = 2, True, k++, If[f[n, k] != {}, Return[k]]];
a /@ Range[2, 200] (* Jean-François Alcover, Oct 03 2020 *)

Prog

(PARI) A161882(n)={vecmin(factor(n)[, 1]%4)==1 && return(2);  if(n==1<<valuation(n, 2), 4, 3)} \\ M. F. Hasler, Dec 17 2014

Crossrefs

Cf. A161883, A161884, A161885.

Sequence in context: A215597 A266110 A204819 * A276789 A082125 A058290

Adjacent sequences: A161879 A161880 A161881 * A161883 A161884 A161885

Keyword

nonn

Author

Dmitry Kamenetsky, Jun 21 2009

Extensions

More terms from Alois P. Heinz, Dec 04 2014

Status

approved