OFFSET
1,1
COMMENTS
The sequence is likely to be finite and complete as the next term, if it exists, is > 50000.
From a proof by David A. Corneth on Nov 08 2017 in A294736: This sequence is complete, see the von Eitzen Link and Price's computation that the next term must be > 50000. Proof. The link mentions "for positive integer n, if n > 5408 then the number of ways to write n as a sum of 5 squares is at least Floor(Sqrt(n - 101) / 8)". So for n > 5408, there are more than one way to write n as a sum of 5 squares. For n <= 5408, it has been verified if n is in the sequence by inspection. Hence the sequence is complete.
REFERENCES
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
LINKS
H. von Eitzen, in reply to user James47, What is the largest integer with only one representation as a sum of five nonzero squares? on stackexchange.com, May 2014
D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
Eric Weisstein's World of Mathematics, Square Number.
FORMULA
A243148(a(n),5) = 1. - Alois P. Heinz, Feb 25 2019
MATHEMATICA
Select[Range[100], Length[Select[PowersRepresentations[#, 5, 2], #[[1]] > 0&]] == 1&] (* Jean-François Alcover, Feb 25 2019 *)
b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i<1 || t<1, 0, b[n, i - 1, k, t] + If[i^2 > n, 0, b[n - i^2, i, k, t - 1]]]];
T[n_, k_] := b[n, Sqrt[n] // Floor, k, k];
Position[Table[T[n, 5], {n, 0, 100}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Robert Price, Nov 06 2017
STATUS
approved