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A343397
Number of ways to write n as 2^x + [y^2/3] + [z^2/4] with x,y,z positive integers, where [.] is the floor function.
10
0, 1, 2, 3, 4, 4, 5, 5, 8, 5, 9, 5, 8, 8, 6, 9, 9, 10, 8, 11, 10, 10, 9, 9, 14, 8, 8, 10, 12, 11, 6, 14, 13, 10, 12, 13, 15, 11, 13, 9, 20, 6, 12, 17, 13, 13, 10, 11, 17, 12, 11, 13, 15, 14, 9, 13, 13, 14, 11, 18, 11, 15, 7, 12, 22, 13, 14, 17, 17, 11, 15, 13, 24, 16, 9, 17, 15, 15, 14, 18
OFFSET
1,3
COMMENTS
Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for all n = 2..2*10^6.
Conjecture verified up to 10^10. - Giovanni Resta, Apr 14 2021
LINKS
Zhi-Wei Sun, Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c], arXiv:1504.01608 [math.NT], 2015.
EXAMPLE
a(2) = 1 with 2 = 2^1 + [1^2/3] + [1^2/4].
a(3) = 2 with 3 = 2^1 + [1^2/3] + [2^2/4] = 2^1 + [2^2/3] + [1^2/4].
MATHEMATICA
PowQ[n_]:=PowQ[n]=n>1&&IntegerQ[Log[2, n]];
tab={}; Do[r=0; Do[If[PowQ[n-Floor[x^2/3]-Floor[y^2/4]], r=r+1], {x, 1, Sqrt[3n-1]}, {y, 1, Sqrt[4(n-Floor[x^2/3]-1)+1]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 13 2021
STATUS
approved