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A180149 Integers with precisely two partitions into sums of four squares of nonnegative numbers.

%I A180149 #19 Feb 01 2021 02:22:31
%S A180149 4,9,10,12,13,16,17,19,20,21,22,29,30,31,35,39,40,44,46,47,48,64,71,
%T A180149 80,88,120,160,176,184,192,256,320,352,480,640,704,736,768,1024,1280,
%U A180149 1408,1920,2560,2816,2944,3072,4096,5120,5632,7680
%N A180149 Integers with precisely two partitions into sums of four squares of nonnegative numbers.
%C A180149 The largest odd member of this sequence is 71, and from a(32)=320 onwards the terms satisfy the eighth-order recurrence relation a(n)=4a(n-8).
%C A180149 A002635(a(n)) = 2. - _Reinhard Zumkeller_, Jul 13 2014
%H A180149 Robert Price, <a href="/A180149/b180149.txt">Table of n, a(n) for n = 1..65</a>
%H A180149 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 A180149 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A180149 The members of this sequence are {9, 13, 17, 19, 21, 29, 30, 31, 35, 39, 46, 47, 71} together with all integers of the form 5*2^N, 11*2^N and {1,3,30,46}*4^N where N > 0 (which includes a necessary correction to Lehmer's result).
%e A180149 As the fifth integer which has precisely two partitions into sums of four squares of nonnegative numbers is 13, then a(5)=13.
%t A180149 Select[Range[10000], Length[PowersRepresentations[ #, 4, 2]]==2&]
%o A180149 (Haskell)
%o A180149 a180149 n = a180149_list !! (n-1)
%o A180149 a180149_list = filter ((== 2) . a002635) [0..]
%o A180149 -- _Reinhard Zumkeller_, Jul 13 2014
%Y A180149 Cf. A002635, A006431, A245022.
%K A180149 easy,nonn
%O A180149 1,1
%A A180149 _Ant King_, Aug 17 2010