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

Author

N. J. A. Sloane, Jul 19 2018

Keywords

• nonn
• more

Comments

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

Examples

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

Links

• 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.

Crossrefs

Cf. A316833.

Programs

Maple

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

Mathematica

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 *)