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Numbers that are the sum of 5 nonzero squares in exactly 3 ways.
2

%I #14 Dec 03 2017 00:56:38

%S 29,32,35,37,40,43,44,46,51,52,58,65,69,73,78,87,90

%N Numbers that are the sum of 5 nonzero squares in exactly 3 ways.

%C This sequence is likely finite and complete as the next term, if it exists, is > 50000.

%C 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 eight 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.

%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

%H H. von Eitzen, in reply to user James47, <a href="http://math.stackexchange.com/questions/811824/what-is-the-largest-integer-with-only-one-representation-as-a-sum-of-five-nonzer">What is the largest integer with only one representation as a sum of five nonzero squares?</a> on stackexchange.com, May 2014

%H D. H. Lehmer, <a href="http://www.jstor.org/stable/2305380">On the Partition of Numbers into Squares</a>, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number.</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%Y Cf. A025429, A025357, A294675, A294736.

%K nonn,fini,full

%O 1,1

%A _Robert Price_, Nov 07 2017