A138554 Minimal value of sum k_i when sum (k_i)^2 = n.

0, 1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 8, 5, 6, 7, 8, 7, 8, 9, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 10, 11, 10, 9, 10, 11, 12, 7, 8, 9, 10, 9, 10, 11, 12, 11, 10, 11, 12, 11, 12, 13, 8, 9, 10, 11, 10, 11, 12, 13, 12, 11, 12, 13, 14, 13, 14, 15, 12, 9, 10, 11, 12, 11

Offset: 0

Author

Franklin T. Adams-Watters, Mar 24 2008

Keywords

• nonn
• look

Examples

			32 = 4^2 + 4^2 and 4+4 = 8. Using 5, the best we can do is 32 = 5^2 + 2^2 + 1^2 + 1^2 + 1^2 and 5+2+1+1+1 = 10, so a(32) = 8.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20163
• Phillip Tomas Heikoop, Dimensions of Matrix Subalgebras, Bachelor's Thesis, Worcester Polytechnic Institute (Massachusetts, 2019).

Crossrefs

Cf A063772, A138555, A001156.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, n,
       min(seq(b(n-j*i^2, i-1)+j*i, j=0..n/i^2)))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 30 2015

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, n, Min[Table[b[n - j i^2, i - 1] + j i, {j, 0, n/i^2}]]];
a[n_] := b[n, Sqrt[n] // Floor];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

PARI

sslist(n) = {local(r,i,v,t); r=vector(n+1,k,0); for(k=1,n,v=k;i=1;while(i^2<=k,t=r[k-i^2+1]+i;if(t