cp's OEIS Frontend

A294736 Numbers that are the sum of 5 nonzero squares in exactly 2 ways.

%I A294736 #28 Feb 16 2025 08:33:51
%S A294736 20,38,41,45,47,48,49,50,54,55,63,66,81,105
%N A294736 Numbers that are the sum of 5 nonzero squares in exactly 2 ways.
%C A294736 Inspected values of n <= 50000.
%C 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 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
%D A294736 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
%H A294736 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 A294736 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 A294736 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A294736 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A294736 A243148(a(n),5) = 2. - _Alois P. Heinz_, Feb 26 2019
%e A294736 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.
%t A294736 Select[Range[200], Length[Select[PowersRepresentations[#, 5, 2], #[[1]] > 0&]] == 2&] (* _Jean-François Alcover_, Nov 06 2020 *)
%Y A294736 Cf. A025429, A025357, A243148, A294675.
%K A294736 nonn,fini,full
%O A294736 1,1
%A A294736 _Robert Price_, Nov 07 2017