A347865 Number of ways to write n as w^2 + 2*x^2 + y^4 + 3*z^4, where w,x,y,z are nonnegative integers.

1, 2, 2, 3, 4, 3, 3, 3, 2, 4, 3, 2, 5, 3, 1, 2, 3, 3, 4, 6, 5, 4, 6, 3, 2, 6, 2, 5, 7, 1, 3, 3, 2, 4, 5, 4, 6, 7, 4, 3, 3, 4, 2, 4, 4, 2, 3, 2, 4, 6, 5, 7, 10, 4, 7, 7, 1, 9, 6, 3, 7, 3, 2, 2, 4, 5, 7, 11, 6, 4, 9, 3, 5, 11, 2, 7, 10, 2, 2, 2, 4, 8, 12, 7, 9, 10, 7, 6, 5, 7, 6, 7, 8, 5, 1, 2, 4, 10, 7, 11, 15

Offset

0,2

Comments

Conjecture 1: a(n) > 0 except for n = 744.

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

It seems that a(n) = 1 only for n = 0, 14, 29, 56, 94, 110, 158, 159, 224, 239, 296, 464, 589, 1214, 1454, 1709.

Conjecture 2: For any positive odd integer a, all sufficiently large integers can be written as a*w^4 + 2*x^4 + (2*y)^2 + z^2 with w,x,y,z integers. If M(a) denotes the largest integer not of the form a*w^4 + 2*x^4 + (2*y)^2 + z^2 (with w,x,y,z integers), then M(1) = 255, M(3) = 303, M(5) = 497, M(7) = 3182, M(9) = 4748, M(11) = 5662, M(13) = 5982, M(15) = 10526, M(17) = 4028 and M(19) = 11934.

Conjecture 3: Let E(a,b,c) be the set of nonnegative integers not of the form w^2 + a*x^2 + b*y^4 + c*z^4 with w,x,y,z integers. Then E(1,2,4) = {135, 190, 510}, E(1,2,5) = {35, 254, 334}, E(2,1,4) = {190, 270, 590} and E(2,3,7) = {94, 490, 983} and E(3,1,2) = {56, 168, 378}.

See also A346643 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(14) = 1 with 14 = 3^2 + 2*1^2 + 0^4 + 3*1^4.
a(158) = 1 with 158 = 11^2 + 2*3^2 + 2^4 + 3*1^4.
a(589) = 1 with 589 = 14^2 + 2*14^2 + 1^4 + 3*0^4.
a(1214) = 1 with 1214 = 27^2 + 2*11^2 + 0^4 + 3*3^4.
a(1454) = 1 with 1454 = 27^2 + 2*19^2 + 0^4 + 3*1^4.
a(1709) = 1 with 1709 = 29^2 + 2*0^2 + 5^4 + 3*3^4.

Mathematica

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

Crossrefs

Cf. A000290, A000583, A346643, A350857.

Sequence in context: A307981 A262824 A242899 * A306606 A229989 A126688

Adjacent sequences: A347862 A347863 A347864 * A347866 A347867 A347868

Keyword

nonn

Author

Zhi-Wei Sun, Jan 24 2022

Status

approved