A346643 Number of ways to write n as w^2 + 2*x^2 + 3*y^4 + 4*z^4, where w,x,y,z are nonnegative integers.

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

Offset

0,4

Comments

1-2-3-4 Conjecture: a(n) > 0 except for n = 158.

This has been verified for n up to 10^8.

It seems that a(n) = 1 only for n = 0, 1, 2, 14, 17, 35, 42, 46, 62, 87, 119, 122, 168, 189, 206, 234, 237, 302, 317, 398, 545, 1037, 1437, 4254.

See also A347865 and A350857 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(46) = 1 with 46 = 5^2 + 2*3^2 + 3*1^4 + 4*0^4.
a(119) = 1 with 119 = 7^2 + 2*3^2 + 3*2^4 + 4*1^4.
a(398) = 1 with 398 = 13^2 + 2*9^2 + 3*1^4 + 4*2^4.
a(545) = 1 with 545 = 19^2 + 2*6^2 + 3*2^4 + 4*2^4.
a(1037) = 1 with 1037 = 31^2 + 2*6^2 + 3*0^4 + 4*1^4.
a(1437) = 1 with 1437 = 9^2 + 2*26^2 + 3*0^4 + 4*1^4.
a(4254) = 1 with 4254 = 45^2 + 2*31^2 + 3*3^4 + 4*2^4.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-4x^4-3y^4-2z^2], r=r+1], {x, 0, (n/4)^(1/4)}, {y, 0, ((n-4x^4)/3)^(1/4)}, {z, 0, Sqrt[(n-4x^4-3y^4)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000583, A347865, A350857.

Sequence in context: A026263 A080098 A083060 * A272979 A357610 A102351

Adjacent sequences: A346640 A346641 A346642 * A346644 A346645 A346646

Keyword

nonn

Author

Zhi-Wei Sun, Jan 24 2022

Status

approved