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