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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, 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
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
KEYWORD
nonn,look
AUTHOR
Moshe Shmuel Newman, May 15 2014
STATUS
approved