A241898 a(n) is the largest integer such that n = a(n)^2 + ... is a decomposition of n into a sum of at most four nondecreasing squares.

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

Offset: 1

Author

Moshe Shmuel Newman, May 15 2014

Keywords

• nonn
• look

Comments

This differs from A191090 only for n>=30 because 30 cannot be written as a sum of at most four squares without using 1^2, but 30 can be written as a sum of five nondecreasing squares: 2^2 + 2^2 + 2^2 + 3^2 + 3^2, making A191090(30)=2.

By Lagrange's Theorem every number can be written as a sum of four squares. Can the same be said of the set of {a^2|a is any integer not equal to 7}? From the data that I have, it would seem that a(n) is greater than 7 for all n>599. If this could be proved, it would only remain to check if all the numbers up to 599 can be written as the sum of 4 squares none of which is 7^2.

Examples

			30 can be written as the sum of at most 4 nondecreasing squares in the following ways: 1^2 + 2^2 + 5^2 or 1^2 + 2^2 + 3^2 + 4^2. Therefore, a(30)=1.
		

Links

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

Crossrefs

Cf. A191090.

Programs

Maple

b:= proc(n, i, t) option remember; n=0 or t>0 and
       i^2<=n and (b(n, i+1, t) or b(n-i^2, i, t-1))
    end:
a:= proc(n) local k;
      for k from isqrt(n) by -1 do
        if b(n, k, 4) then return k fi
      od
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, May 25 2014

Mathematica

For[i=0,i<=7^4,i++,a[i]={}];
For[i1=0,i1<=7,i1++,
For[i2=0,i2<=7,i2++,
For[i3=0,i3<=7,i3++,
For[i4=0,i4<=7,i4++,
sumOfSquares=i1^2+i2^2+i3^2+i4^2;
smallestSquare=Min[DeleteCases[{i1,i2,i3,i4},0]];
a[sumOfSquares]=Union[{smallestSquare},a[sumOfSquares]] ]]]];
Table[Max[a[i]],{i,1,50}]