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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.
6
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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 24 2022
STATUS
approved