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%I #40 Feb 01 2022 00:57:09
%S 0,25,325,1105,4225,5525,203125,27625,71825,138125,2640625,160225,
%T 17850625,1221025,1795625,801125,1650390625,2082925,49591064453125,
%U 4005625,44890625,2158203125,30525625,5928325,303460625,53955078125
%N Smallest number that is the sum of 2 squares (allowing zeros) in exactly n ways.
%H Ray Chandler, <a href="/A000446/b000446.txt">Table of n, a(n) for n = 1..1458</a> (a(1459) exceeds 1000 digits).
%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F An algorithm to compute the n-th term of this sequence for n>1: Write each of 2n and 2n-1 as products of their divisors, in decreasing order and in all possible ways. Equate each divisor in the product to (a1+1)(a2+1)...(ar+1), so that a1>=a2>=a3>=...>=ar, and solve for the ai. Evaluate A002144(1)^a1 x A002144(2)^a2 x ... x A002144(r)^ar for each set of values determined above, then the smaller of these products is the least integer to have precisely n partitions into a sum of two squares. [_Ant King_, Oct 07 2010]
%F a(n) = min(A018782(2n-1),A018782(2n)) for n>1.
%e a(1) = 0 because 0 is the smallest integer which is uniquely a unique sum of two squares, namely 0^2 + 0^2.
%e a(2) = 25 from 25 = 5^2 + 0^2 = 3^2 + 4^2.
%e a(3) = 325 from 325 = 1^2 + 18^2 = 6^2 + 17^2 = 10^2 + 15^2.
%e a(4) = 1105 from 1105 = 4^2 + 33^2 = 9^2 + 32^2 = 12^2 + 31^2 = 23^2 + 24^2.
%Y Cf. A002144, A018782, A054994.
%Y See A016032, A093195 and A124980 for other versions.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E Better description and more terms from _David W. Wilson_, Aug 15 1996
%E Definition improved by several correspondents, Nov 12 2007
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