

A281975


Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and xy are squares.


7



1, 3, 4, 3, 2, 4, 4, 2, 1, 4, 5, 3, 1, 2, 3, 2, 3, 5, 7, 3, 4, 5, 2, 1, 2, 6, 9, 6, 2, 4, 6, 3, 4, 6, 8, 4, 6, 5, 4, 3, 2, 11, 10, 4, 1, 7, 5, 1, 3, 3, 11, 9, 7, 5, 6, 4, 2, 7, 5, 4, 2, 6, 4, 3, 2, 7, 15, 4, 4, 6, 5, 1, 2, 6, 7, 7, 3, 6, 4, 2, 4
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OFFSET

0,2


COMMENTS

Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
(ii) Each nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that xy and 2*(yz) (or 2*(zy)) are both squares.
(iii) For each ordered pair (a,b) = (2,1), (3,1), (9,5), (14,10), any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x and a*xb*y are both squares.
The author has proved that each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x (or xy, or 2(xy)) is a square.
See also A281976 and A281977 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.


EXAMPLE

a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and 00 = 0^2.
a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 = 1^2 and 11 = 0^2.
a(44) = 1 since 44 = 1^2 + 5^2 + 3^2 + 3^2 with 1 = 1^2 and 15 = 2^2.
a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 = 1^2 and 11 = 0^2.
a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 = 1^2 and 15 = 2^2.
a(95) = 1 since 95 = 1^2 + 2^2 + 3^2 + 9^2 with 1 = 1^2 and 12 = 1^2.
a(140) = 1 since 140 = 9^2 + 5^2 + 3^2 + 5^2 with 9 = 3^2 and 95 = 2^2.
a(428) = 1 since 428 = 9^2 + 13^2 + 3^2 + 13^2 with 9 = 3^2 and 913 = 2^2.
a(568) = 1 since 568 = 4^2 + 8^2 + 2^2 + 22^2 with 4 = 2^2 and 48 = 2^2.
a(632) = 1 since 632 = 16^2 + 12^2 + 6^2 + 14^2 with 16 = 4^2 and 1612 = 2^2.
a(1144) = 1 since 1144 = 16^2 + 20^2 + 2^2 + 22^2 with 16 = 4^2 and 1620 = 2^2.
a(1544) = 1 since 1544 = 0^2 + 0^2 + 10^2 + 38^2 with 0 = 0^2 and 00 = 0^2.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[nx^4y^2z^2]&&SQ[Abs[x^2y]], r=r+1], {x, 0, n^(1/4)}, {y, 0, Sqrt[nx^4]}, {z, 0, Sqrt[(nx^4y^2)/2]}]; Print[n, " ", r]; Continue, {n, 0, 80}]


CROSSREFS

Cf. A000118, A000290, A270969, A271775, A281939, A281941, A281976, A281977.
Sequence in context: A258451 A164358 A275638 * A133617 A199286 A188722
Adjacent sequences: A281972 A281973 A281974 * A281976 A281977 A281978


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 03 2017


STATUS

approved



