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A328151 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + y^2 = k.

%I A328151 #27 Jun 29 2024 10:49:25
%S A328151 0,2,5,50,65,1250,325,31250,1105,8450,8125,19531250,5525,488281250,
%T A328151 105625,211250,27625,305175781250,71825,7629394531250,138125,5281250,
%U A328151 126953125,4768371582031250,160225,35701250,1221025,2442050,3453125
%N A328151 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + y^2 = k.
%C A328151 a(n) is the smallest nonnegative i such that A063725(i) = n.
%C A328151 If a(n) exists, then a(n) is of the form 2*m^2 if and only if n is odd. - _Chai Wah Wu_, Jun 28 2024
%H A328151 Chai Wah Wu, <a href="/A328151/a328151.txt">terms a(n) for n odd and a(n) <= 3433037534088061250.</a>
%F A328151 Conjecture: a(2k) = A093195(k) for k >= 1, a(2k+1) = 2*A006339(k)^2 for k >= 0. - _Jon E. Schoenfield_, Jan 23 2022
%e A328151 For n = 3: The sums of the two members of each of the pairs (1, 49), (25, 25) and (49, 1) is 50 and 50 is the smallest nonnegative integer where exactly 3 such pairs exist, so a(3) = 50.
%o A328151 (PARI) a063725(n) = if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ after _Charles R Greathouse IV_ in A063725
%o A328151 a(n) = for(x=0, oo, if(a063725(x)==n, return(x)))
%o A328151 (Python)
%o A328151 # uses Python code from A063725
%o A328151 from itertools import count
%o A328151 def A328151(n): return next(m for m in ((k**2<<1) if n&1 else k for k in count(0)) if A063725(m)==n) # _Chai Wah Wu_, Jun 28 2024
%Y A328151 Cf. A063725, A093195.
%K A328151 nonn,hard,more
%O A328151 0,2
%A A328151 _Felix Fröhlich_, Oct 05 2019
%E A328151 a(13)-a(22) from _Bert Dobbelaere_, Oct 20 2019
%E A328151 a(23)-a(28) from _Chai Wah Wu_, Jun 28 2024