A295670 Numbers that have exactly one representation as a sum of six positive squares.

6, 9, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 27, 28, 31, 32, 34, 35, 37, 40, 43

Offset: 1

Author

Robert Price, Nov 25 2017

Keywords

• nonn
• more

Comments

It appears that this sequence is finite and complete. See the von Eitzen link for a proof for the 5 positive squares case.

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.

Crossrefs

Cf. A000177, A025430, A243148, A294524.

Programs

Mathematica

m = 6;
r[n_] := Reduce[xx = Array[x, m]; 0 <= x[1] && LessEqual @@ xx && AllTrue[xx, Positive] && n == Total[xx^2], xx, Integers];
For[n = 0, n < 50, n++, rn = r[n]; If[rn[[0]] === And, Print[n, " ", rn]]] (* 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, 6], {n, 0, 100}], 1] - 1 // Flatten (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz in A243148 *)

Formula

A243148(a(n),6) = 1. - Alois P. Heinz, Feb 25 2019