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%I #29 Nov 04 2020 05:23:48
%S 84,116,140,164,180,196,212,236,244,332,460,628
%N Numbers with a unique representation as a sum of four distinct odd squares.
%C Numbers n that have a unique representation as n = h^2 + i^2 + j^2 + k^2 with h,i,j,k odd and 0 < h < i < j < k.
%C No more terms up to 5*10^5. - _Robert Israel_, Jul 20 2018
%C a(13) > 5*10^6, if it exists. - _Robert Price_, Jul 25 2018
%C a(13) > 10^11, if it exists (which seems very unlikely). - _Jon E. Schoenfield_, Jul 28 2018
%H Michael D. Hirschhorn, <a href="https://ajc.maths.uq.edu.au/pdf/24/ocr-ajc-v24-p285.pdf">Partitions into Four Distinct Squares of Equal Parity</a>, Australasian Journal of Combinatorics, Volume 24(2001), pp. 285-291.
%H Michael D. Hirschhorn, <a href="https://doi.org/10.1007/978-3-319-57762-3">The Power of q: A Personal Journey</a>, Springer 2017. See Chapter 31: Partitions into Four Distinct Squares of Equal Parity.
%e 156 (a member of A316833) is not a member here since it has two representations: 156 = 1+25+49+81 = 1+9+25+121.
%p N:= 10000: # to get all terms <= N
%p V:= Vector(N):
%p for a from 1 to floor(sqrt(N/4)) by 2 do
%p for b from a+2 to floor(sqrt((N-a^2)/3)) by 2 do
%p for c from b+2 to floor(sqrt((N-a^2-b^2)/2)) by 2 do
%p for d from c + 2 by 2 do
%p r:= a^2+b^2+c^2+d^2;
%p if r > N then break fi;
%p V[r]:= V[r]+1
%p od od od od:
%p select(r -> V[r]=1, [$1..N]); # _Robert Israel_, Jul 20 2018
%t okQ[n_] := Count[PowersRepresentations[n, 4, 2], pr_List /; Union[pr] == pr && AllTrue[pr, OddQ]] == 1;
%t Select[Range[1000], okQ] (* _Jean-François Alcover_, Apr 02 2019 *)
%Y Cf. A316833.
%K nonn,more
%O 1,1
%A _N. J. A. Sloane_, Jul 19 2018
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