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

1,1

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

Table of n, a(n) for n=1..21.

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.

Formula

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

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 *)

Crossrefs

Cf. A000177, A025430, A243148, A294524.

Sequence in context: A185179 A072546 A344805 * A272466 A267918 A330703

Adjacent sequences: A295667 A295668 A295669 * A295671 A295672 A295673

Keyword

nonn,more

Author

Robert Price, Nov 25 2017

Status

approved