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A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.

%I A016032 #49 Feb 16 2025 08:32:33
%S A016032 2,50,325,1105,8125,5525,105625,27625,71825,138125,5281250,160225,
%T A016032 1221025,2442050,1795625,801125,446265625,2082925,41259765625,4005625,
%U A016032 44890625,30525625,61051250,5928325,303460625,53955078125,35409725,100140625,1289367675781250
%N A016032 Least positive integer that is the sum of two squares of positive integers in exactly n ways.
%D A016032 A. Beiler, Recreations in the Theory of Numbers, Dover, pp. 140-141.
%H A016032 T. D. Noe and Ray Chandler, <a href="/A016032/b016032.txt">Table of n, a(n) for n = 1..2178</a> (a(2179) exceeds 1000 digits).
%H A016032 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_062.htm">Puzzle 62</a>
%H A016032 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H A016032 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~twosquares.en.html">Two squares</a>
%H A016032 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A016032 a(n) = min(2*A018782(2n-1), A018782(2n), A018782(2n+1)).
%e A016032 a(0) = 1 as 1 is the least positive integer not expressible as the sum of two squared positives.
%e A016032 a(1) = 2 from 2 = 1^2 + 1^2.
%e A016032 a(2) = 50 from 50 = 1^2 + 7^2 = 5^2 + 5^2.
%t A016032 Array[Block[{k = 1}, While[Length@ DeleteCases[PowersRepresentations[k, 2, 2], _?(! FreeQ[#, 0] &)] != #, k++]; k] &, 6] (* _Michael De Vlieger_, Mar 31 2019 *)
%o A016032 (PARI) b(k)=my(c=0);for(i=1,sqrtint(k\2),if(issquare(k-i^2),c+=1));c \\ A025426
%o A016032 for(n=1,10,k=1;while(k,if(b(k)==n,print1(k,", ");break);k+=1)) \\ _Derek Orr_, Mar 20 2019
%Y A016032 Cf. A018825, A048610, A025284-A025293 (first entries).
%Y A016032 See A000446, A124980 and A093195  for other versions.
%K A016032 nonn,nice
%O A016032 1,1
%A A016032 _Robert G. Wilson v_
%E A016032 Corrected and extended by _Jud McCranie_
%E A016032 Definition improved by several correspondents, Nov 12 2007