A350088 Number of ways to write n as t^2 + w^4 + 2*x^4 + 4*y^4 + 8*z^4, where t is a positive integer, and w,x,y,z are nonnegative integers.

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 4, 3, 3, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 3, 3, 4, 2, 4, 5, 4, 3, 4, 5, 4, 2, 3, 5, 4, 2, 3, 7, 3, 3, 4, 5, 3, 2, 4, 5, 3, 1, 4, 5, 3, 1, 5, 5, 4, 3, 5, 6, 3, 3, 3, 6, 3, 4, 6, 5, 2, 3, 3, 7, 6, 4, 6, 4, 5, 2, 5, 7, 7, 6, 7, 3, 6, 2, 6, 8, 4, 5, 6

Offset

1,4

Comments

Conjecture: If (a,b,c) is among the triples (2,3,4), (2,3,8), (2,4,6), (2,4,7), (2,4,8) and (2,4,11), then any positive integer n can be written as t^2 + w^4 + a*x^4 + b*y^4 + c*z^4, where t is a positive integer, and w,x,y,z are nonnegative integers.

We have verified this for all n = 1..10^8.

It seems that a(n) = 1 only for n = 1, 2, 3, 59, 63, 119, 223, 1087.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

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

Example

a(1) = 1 with 1 = 1^2 + 0^4 + 2*0^4 + 4*0^4 + 8*0^4.
a(2) = 1 with 2 = 1^2 + 1^4 + 2*0^4 + 4*0^4 + 8*0^4.
a(3) = 1 with 3 = 1^2 + 0^4 + 2*1^4 + 4*0^4 + 8*0^4.
a(59) = 1 with 59 = 7^2 + 0^4 + 2*1^4 + 4*0^4 + 8*1^4.
a(63) = 1 with 63 = 7^2 + 0^4 + 2*1^4 + 4*1^4 + 8*1^4.
a(119) = 1 with 119 = 6^2 + 3^4 + 2*1^4 + 4*0^4 + 8*0^4.
a(223) = 1 with 223 = 7^2 + 0^4 + 2*3^4 + 4*1^4 + 8*1^4.
a(1087) = 1 with 1087 = 14^2 + 3^4 + 2*3^4 + 4*0^4 + 8*3^4.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-w^4-2x^4-4*y^4-8z^4], r=r+1], {w, 0, (n-1)^(1/4)}, {x, 0, ((n-1-w^4)/2)^(1/4)}, {y, 0, ((n-1-w^4-2x^4)/4)^(1/4)}, {z, 0, ((n-1-w^4-2x^4-4y^4)/8)^(1/4)}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000583, A349246.

Sequence in context: A204166 A227581 A263846 * A178786 A350174 A000196

Adjacent sequences: A350085 A350086 A350087 * A350089 A350090 A350091

Keyword

nonn

Author

Zhi-Wei Sun, Mar 26 2022

Status

approved