login
The OEIS is supported by the many generous donors to the OEIS 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 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A350857 Number of ways to write n as 2*w^4 + x^4 + y^2 + z^2, where w,x,y,z are nonnegative integers with w or x a square. 7
1, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 1, 1, 3, 3, 5, 4, 5, 3, 2, 2, 1, 3, 5, 4, 4, 3, 1, 2, 4, 4, 6, 4, 5, 5, 4, 2, 3, 6, 5, 5, 2, 3, 2, 2, 3, 3, 7, 4, 7, 6, 3, 3, 2, 3, 5, 3, 1, 4, 2, 2, 3, 5, 6, 5, 7, 4, 3, 2, 2, 4, 5, 3, 3, 3, 1, 1, 2, 6, 8, 9, 5, 7, 5, 3, 4, 3, 6, 6, 4, 3, 2, 0, 3, 6, 7, 5, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Conjecture 1: a(n) > 0 except for n = 95, 255.

This has been verified for n up to 10^7.

It seems that a(n) = 1 only for n = 0, 14, 15, 24, 30, 60, 78, 79, 111, 159, 174, 249, 270, 303, 318, 334, 382, 399, 414, 462, 472, 831, 894, 975, 1080, 1151, 1164, 1919, 4080, 6456, 20319.

Conjecture 2: For any positive integer a == 2 (mod 4), all sufficiently large integers can be written as a*w^4 + x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If N(a) denotes the largest integer not of the form a*w^4 + x^4 + (2*y)^2 + z^2 (w,x,y,z = 0,1,2,...), then we have N(2) = 255, N(6) = 2716, N(10) = 598, N(14) = 8427, N(18) = 2463, N(22) = 3884, N(26) = 14988, N(30) = 10843.

See also A346643 and A347865 for similar conjectures.

LINKS

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

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97-120.

Zhi-Wei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 2020-2022.

EXAMPLE

a(24) = 1 with 24 = 2*0^4 + 2^4 + 2^2 + 2^2 and 0 = 0^2.

a(1151) = 1 with 1151 = 2*3^4 + 4^4 + 2^2 + 27^2 and 4 = 2^2.

a(6456) = 1 with 6456 = 2*1^4 + 3^4 + 17^2 + 78^2 and 1 = 1^2.

a(20319) = 1 with 20319 = 2*5^4 + 0^4 + 5^2 + 138^2 and 0 = 0^2.

MATHEMATICA

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

tab={}; Do[r=0; Do[If[SQ[n-2w^4-x^4-y^2]&&(SQ[w]||SQ[x]), r=r+1], {w, 0, (n/2)^(1/4)}, {x, 0, (n-2w^4)^(1/4)}, {y, 0, Sqrt[(n-2w^4-x^4)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

CROSSREFS

Cf. A000290, A000583, A346643, A347865.

Sequence in context: A358617 A008968 A162499 * A135715 A089326 A237367

Adjacent sequences: A350854 A350855 A350856 * A350858 A350859 A350860

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 19 2022

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 19:33 EST 2022. Contains 358648 sequences. (Running on oeis4.)