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A316834 Numbers with a unique representation as a sum of four distinct odd squares. 5
84, 116, 140, 164, 180, 196, 212, 236, 244, 332, 460, 628 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..12.

Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31: Partitions into Four Distinct Squares of Equal Parity.

EXAMPLE

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

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-Fran├žois Alcover, Apr 02 2019 *)

CROSSREFS

Cf. A316833.

Sequence in context: A209204 A219801 A316833 * A227734 A192322 A015708

Adjacent sequences:  A316831 A316832 A316833 * A316835 A316836 A316837

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Jul 19 2018

STATUS

approved

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Last modified June 1 20:49 EDT 2020. Contains 334765 sequences. (Running on oeis4.)