

A294736


Numbers that are the sum of 5 nonzero squares in exactly 2 ways.


9



20, 38, 41, 45, 47, 48, 49, 50, 54, 55, 63, 66, 81, 105
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OFFSET

1,1


COMMENTS

Inspected values of n <= 50000.
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 two ways 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."  David A. Corneth, Nov 08 2017


REFERENCES

E. Grosswald, Representations of Integers as Sums of Squares. SpringerVerlag, New York, 1985, p. 86, Theorem 1.


LINKS

Table of n, a(n) for n=1..14.
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. 476481.
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares


FORMULA

A243148(a(n),5) = 2.  Alois P. Heinz, Feb 26 2019


EXAMPLE

There are exactly two ways 20 is a sum of 5 nonzero squares. These are 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 = 20. Therefore 20 is in the sequence.


MATHEMATICA

Select[Range[200], Length[Select[PowersRepresentations[#, 5, 2], #[[1]] > 0&]] == 2&] (* JeanFrançois Alcover, Nov 06 2020 *)


CROSSREFS

Cf. A025429, A025357, A243148, A294675.
Sequence in context: A308664 A184065 A259734 * A217427 A165442 A242532
Adjacent sequences: A294733 A294734 A294735 * A294737 A294738 A294739


KEYWORD

nonn,fini,full


AUTHOR

Robert Price, Nov 07 2017


STATUS

approved



