login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A338272 Number of ways to write 4*n + 1 as x^2 + y^2 + z^2 + w^2 (0 <= x <= y and 0 <= z <= w) such that x*y + 32*z*w is a square. 1
2, 3, 4, 2, 4, 4, 4, 3, 3, 3, 4, 5, 4, 3, 3, 3, 5, 4, 4, 2, 8, 6, 6, 2, 4, 6, 4, 4, 5, 4, 5, 6, 5, 1, 3, 4, 6, 6, 6, 4, 6, 3, 6, 2, 3, 4, 5, 8, 4, 3, 6, 5, 5, 3, 2, 4, 8, 4, 3, 3, 5, 6, 4, 2, 5, 10, 6, 5, 4, 3, 6, 4, 7, 5, 8, 2, 6, 6, 4, 3, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Conjecture: Any positive integer congruent to 1 modulo 4 can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that a*x*y + b*z*w is a square, provided that (a,b) is among the following ordered pairs: (1,-2), (1,2), (1,10), (1,32), (2,-1), (2,10), (2,14), (2,16), (2,36), (3,4), (4,-2), (4,18), (6,18), (8,9), (9,10), (16,-2).

a(n) > 0 for all n = 0..10^5.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190. See also arXiv:1604.06723 [math.NT].

Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893.  See also arXiv:1701.05868 [math.NT].

Zhi-Wei Sun, Sums of four squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020.

EXAMPLE

a(33) = 1, and 4*33 + 1 = 4^2 + 9^2 + 0^2 + 6^2 with 4*9 + 32*0*6 = 6^2.

a(364) = 1, and 4*364 + 1 = 16^2 + 25^2 + 0^2 + 24^2 with 16*25 + 32*0*24 = 20^2.

a(1319) = 1, and 4*1319 + 1 = 20^2 + 36^2 + 10^2 + 59^2 with 20*36 + 32*10*59 = 140^2.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

tab={}; Do[r=0; Do[If[SQ[4n+1-x^2-y^2-z^2]&&SQ[x*y+32*z*Sqrt[4n+1-x^2-y^2-z^2]], r=r+1],

{x, 0, Sqrt[2n]}, {y, x, Sqrt[4n+1-x^2]}, {z, 0, Sqrt[(4n+1-x^2-y^2)/2]}]; tab=Append[tab, r], {n, 0, 80}]; tab

CROSSREFS

Cf. A000118, A000290, A270073.

Sequence in context: A107468 A023632 A286844 * A107572 A043264 A268444

Adjacent sequences:  A338269 A338270 A338271 * A338273 A338274 A338275

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 19 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 07:17 EST 2021. Contains 349563 sequences. (Running on oeis4.)