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

1,4

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.

Links

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

Example

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.

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}]

Crossrefs

Cf. A191090.

Sequence in context: A162401 A324924 A301314 * A191090 A331958 A319193

Adjacent sequences: A241895 A241896 A241897 * A241899 A241900 A241901

Keyword

nonn,look

Author

Moshe Shmuel Newman, May 15 2014

Status

approved