A241898 - OEIS

%I #48 May 25 2014 19:13:46

%S 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,

%T 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,

%U 2,3,2,6,4,5,5,3,4,2,3,4,9,4,3,4,6,5,2

%N 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.

%C 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.

%C 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.

%H Alois P. Heinz, <a href="/A241898/b241898.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%p b:= proc(n, i, t) option remember; n=0 or t>0 and

%p        i^2<=n and (b(n, i+1, t) or b(n-i^2, i, t-1))

%p     end:

%p a:= proc(n) local k;

%p       for k from isqrt(n) by -1 do

%p         if b(n, k, 4) then return k fi

%p       od

%p     end:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, May 25 2014

%t For[i=0,i<=7^4,i++,a[i]={}];

%t For[i1=0,i1<=7,i1++,

%t For[i2=0,i2<=7,i2++,

%t For[i3=0,i3<=7,i3++,

%t For[i4=0,i4<=7,i4++,

%t sumOfSquares=i1^2+i2^2+i3^2+i4^2;

%t smallestSquare=Min[DeleteCases[{i1,i2,i3,i4},0]];

%t a[sumOfSquares]=Union[{smallestSquare},a[sumOfSquares]] ]]]];

%t Table[Max[a[i]],{i,1,50}]

%Y Cf. A191090.

%K nonn,look

%O 1,4

%A _Moshe Shmuel Newman_, May 15 2014