

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
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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, Partitions into Four Distinct Squares of Equal Parity, Australasian Journal of Combinatorics, Volume 24(2001), pp. 285291.
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((Na^2)/3)) by 2 do
for c from b+2 to floor(sqrt((Na^2b^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] (* JeanFrançois 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



