OFFSET
1,2
COMMENTS
It follows from the formula that there are infinitely many integers that cannot be partitioned into a sum of four distinct squares of nonnegative integers and infinitely many that can. Furthermore, the largest odd number that has no such partition is 103, and thereafter the terms satisfy the thirty-first order recurrence relation a(n) = 4a(n-31). - Ant King, Nov 02 2010
LINKS
Gordon Pall, On Sums of Squares, The American Mathematical Monthly, Vol. 40, No. 1, (January 1933), pp. 10-18. [From Ant King, Nov 02 2010]
FORMULA
Let k>=0. Then the only integers that cannot be partitioned into a sum of four distinct squares of nonnegative integers are 4^k * N3, where N3 = (N1 union N2), and N1 and N2 are defined by N1 = {1,3,5,7,9,11,13,15,17,19,23,25,27,31,33,37,43,47,55,67,73,97,103} and N2 = {2,6,10,18,22,34,58,82}, respectively. - Ant King, Nov 02 2010
MATHEMATICA
data = Reduce[ w^2 + x^2 + y^2 + z^2 == # && 0 <= w < x < y < z < #, {w, x, y, z}, Integers] & /@ Range[112]; DeleteCases[ Table[If[Head[data[[k]]] === Symbol, k, 0], {k, 1, Length[data]}], 0] (* Ant King, Nov 02 2010 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved