A316833 -id:A316833 - cp's OEIS frontend

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

84, 116, 140, 164, 180, 196, 212, 236, 244, 332, 460, 628

Offset: 1

N. J. A. Sloane, Jul 19 2018

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.
No more terms up to 5*10^5. - Robert Israel, Jul 20 2018
a(13) > 5*10^6, if it exists. - Robert Price, Jul 25 2018
a(13) > 10^11, if it exists (which seems very unlikely). - Jon E. Schoenfield, Jul 28 2018

			156 (a member of A316833) is not a member here since it has two representations: 156 = 1+25+49+81 = 1+9+25+121.

Michael D. Hirschhorn, Partitions into Four Distinct Squares of Equal Parity, Australasian Journal of Combinatorics, Volume 24(2001), pp. 285-291.
Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31: Partitions into Four Distinct Squares of Equal Parity.

Cf. A316833.

N:= 10000: # to get all terms <= N
V:= Vector(N):
for a from 1 to floor(sqrt(N/4)) by 2 do
  for b from a+2 to floor(sqrt((N-a^2)/3)) by 2 do
    for c from b+2 to floor(sqrt((N-a^2-b^2)/2)) by 2 do
      for d from c + 2  by 2 do
        r:= a^2+b^2+c^2+d^2;
        if r > N then break fi;
        V[r]:= V[r]+1
od od od od:
select(r -> V[r]=1, [$1..N]); # Robert Israel, Jul 20 2018

okQ[n_] := Count[PowersRepresentations[n, 4, 2], pr_List /; Union[pr] == pr && AllTrue[pr, OddQ]] == 1;
Select[Range[1000], okQ] (* Jean-François Alcover, Apr 02 2019 *)

A316835 Sums of four distinct positive even squares.

Original entry on oeis.org

120, 156, 184, 200, 204, 216, 228, 248, 252, 260, 264, 280, 284, 296, 300, 312, 316, 324, 336, 340, 344, 348, 360, 364, 372, 376, 380, 392, 396, 408, 420, 424, 428, 436, 440, 444, 452, 456, 464, 468, 472, 476, 480, 484, 488, 492, 500, 504, 508, 516, 520, 524, 532, 536, 540, 548, 552

Offset: 1

N. J. A. Sloane, Jul 19 2018

nonn

Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31.

Equals 4*A004433. Cf. A316833, A316834.

Total/@Subsets[(2*Range[10])^2,{4}]//Union (* Harvey P. Dale, May 21 2019 *)

cp's OEIS Frontend

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

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