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 A316834 Numbers with a unique representation as a sum of four distinct odd squares. 5

%I

%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://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, 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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Last modified July 4 15:43 EDT 2020. Contains 335448 sequences. (Running on oeis4.)