A294742 Numbers that are the sum of 5 nonzero squares in exactly 8 ways.

91, 104, 106, 119, 122, 123, 126, 141, 143, 162, 185, 225

Offset: 1

Author

Robert Price, Nov 07 2017

Keywords

• nonn
• fini
• full

Comments

Theorem: There are no further terms. Proof (from a proof by David A. Corneth on Nov 08 2017 in A294736): The von Eitzen link states that 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) = 9. For n <= 5408, terms have been verified by inspection. Hence this sequence is finite and 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.
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A025429, A025357, A294675, A294736.

Programs

Mathematica

fQ[n_] := Block[{pr = PowersRepresentations[n, 5, 2]}, Length@Select[pr, #[[1]] > 0 &] == 8]; Select[Range@250, fQ] (* Robert G. Wilson v, Nov 17 2017 *)