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A282426
Number of ways to write n as x^4 + 4*y^2 + z^2 + 3^k, where x,y,z are nonnegative integers and k is among 0,1,2,3,4.
1
1, 2, 2, 2, 3, 3, 3, 3, 3, 4, 3, 2, 3, 4, 2, 1, 5, 6, 5, 4, 5, 4, 5, 3, 4, 7, 5, 5, 7, 5, 3, 5, 6, 5, 6, 4, 5, 5, 6, 4, 7, 7, 6, 9, 6, 5, 6, 5, 4, 6, 4, 4, 7, 6, 6, 6, 5, 2, 6, 3, 4, 6, 6, 5, 6, 6, 6, 10, 7, 5
OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 0.
(ii) Each n = 1,2,3,... can be written as x^4 + y^2 + z^2 + 2^k with x,y,z nonnegative integers and k among 0,1,2,3,4, 5.
(iii) Any positive integer n can be written as 4*x^4 + y^3 + z^2 + 3^k with k,x,y,z nonnegative integers.
We have verified parts (i) and (ii)-(iii) for n up to 2*10^7 and 10^7 respectively.
See arXiv:1701.05868 for more such conjectures.
LINKS
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.
EXAMPLE
a(1) = 1 since 1 = 0^4 + 4*0^2 + 0^2 + 3^0.
a(16) = 1 since 16 = 0^4 + 4*1^2 + 3^2 + 3^1.
a(475) = 1 since 475 = 3^4 + 4*6^2 + 13^2 + 3^4.
a(556) = 1 since 556 = 0^4 + 4*0^2 + 23^2 + 3^3.
a(8641) = 1 since 8641 = 9^4 + 4*21^2 + 17^2 + 3^3.
a(52696) = 1 since 52696 = 12^4 + 4*87^2 + 41^2 + 3^1.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[n-x^4-4y^2-3^k], r=r+1], {k, 0, Min[4, Log[3, n]]}, {x, 0, (n-3^k)^(1/4)}, {y, 0, Sqrt[(n-3^k-x^4)/4]}]; Print[n, " ", r]; Continue, {n, 1, 80}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 14 2017
STATUS
approved