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A241898
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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.
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1
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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