A316835 -id:A316835 - cp's OEIS frontend

84, 116, 140, 156, 164, 180, 196, 204, 212, 228, 236, 244, 252, 260, 276, 284, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 396, 404, 420, 428, 436, 444, 452, 460, 468, 476, 484, 492, 500, 508, 516, 524, 532, 540, 548, 556, 564, 572, 580, 588, 596, 604, 612, 620, 628, 636, 644, 652, 660

Offset: 1

N. J. A. Sloane, Jul 19 2018

nonn

Theorem (Conjectured by R. William Gosper, proved by M. D. Hirschhorn): Any sum of four distinct odd squares is the sum of four distinct even squares.
The proof uses the following identity:
(4a+1)^2+(4b+1)^2+(4c+1)^2+(4d+1)^2 = 4[ (a+b+c+d+1)^2 + (a-b-c+d)^2 + (a-b+c-d)^2 + (a+b-c-d)^2 ].
All terms == 4 (mod 8).  Are all numbers == 4 (mod 8) and > 412 members of the sequence? - Robert Israel, Jul 20 2018

R. William Gosper and Stephen K. Lucas, Postings to Math Fun Mailing List, July 19 2018
Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31.

Robert Israel, Table of n, a(n) for n = 1..10000

A316834 lists the subsequence for which the representation is unique.

Cf. A004433, A316835.

N:= 1000: # 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(t -> V[t]>=1, [$1..N]); # Robert Israel, Jul 20 2018

cp's OEIS Frontend

A316833 Sums of four distinct odd squares.

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