

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

ZhiWei Sun, Table of n, a(n) for n = 0..10000
ZhiWei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no.2, 97120.
ZhiWei Sun, Sums of four rational squares with certain restrictions, arXiv:2010.05775 [math.NT], 20202022.


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[n2w^4x^4y^2]&&(SQ[w]SQ[x]), r=r+1], {w, 0, (n/2)^(1/4)}, {x, 0, (n2w^4)^(1/4)}, {y, 0, Sqrt[(n2w^4x^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

ZhiWei Sun, Jan 19 2022


STATUS

approved



