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a(n) is the least positive number that can be written in exactly n ways as 3*x*y^2 - x^3 where x and y are positive integers.
1

%I #13 Sep 16 2023 16:45:16

%S 1,2,88,704,36234,285714,2285712,18285696

%N a(n) is the least positive number that can be written in exactly n ways as 3*x*y^2 - x^3 where x and y are positive integers.

%e 1 can't be written as 3*x*y^2 - x^3, so a(0) = 1.

%e 3 * 1 * 1^2 - 1^3 = 2 so a(1) = 2.

%e 3 * 2 * 4^2 - 2^3 = 3 * 8 * 5^2 - 8^3 = 88 so a(2) = 88.

%e 3 * 4 * 8^2 - 4^3 = 3 * 16 * 10^2 - 16^3 = 3 * 64 * 37^2 - 64^3 = 704 so a(3) = 704.

%e 3 * 6 * 45^2 - 6^3 = 3 * 9 * 37^2 - 9^3 = 3 * 33 * 27^2 - 33^3 = 3 * 549 * 317^2 - 549^3 = 36234 so a(4) = 36234.

%e 3 * 9 * 103^2 - 9^3 = 3 * 33 * 57^2 - 33^3 = 3 * 78 * 57^2 - 78^3 = 3 * 333 * 193^2 - 333^3 = 3 * 1221 * 705^2 - 1221^3 = 285714 so a(5) = 285714.

%p N:= 3*10^5: # for terms <= N

%p R:= Vector(N):

%p for x from 1 to floor(N/2) do

%p V:= [seq(3*x*y^2 - x^3, y = ceil(x/sqrt(3)) .. floor(sqrt((x^2+N/x)/3)))];

%p R[V]:= R[V] +~ 1;

%p od:

%p W:= Array(0..max(R)):

%p for i from 1 to N do

%p v:= R[i];

%p if W[v] = 0 then W[v]:= i fi

%p od:

%p convert(W,list);

%t NVal = 2285712; R = ConstantArray[0, NVal]; For[x = 1, x <= Floor[NVal/2], x++, V = Table[3*x*y^2 - x^3, {y, Ceiling[x/Sqrt[3]], Floor[Sqrt[x^2 + NVal/x]/Sqrt[3]]}]; R[[V]] = R[[V]] + 1;]; W = ConstantArray[0, Max[R] + 1]; For[i = 1, i <= NVal, i++, v = R[[i]]; If[W[[v + 1]] == 0, W[[v + 1]] = i];]; W (* _Robert P. P. McKone_, Aug 18 2023 *)

%Y Cf. A135782.

%K nonn,more

%O 0,2

%A _Robert Israel_, Aug 14 2023

%E a(6)-a(7) from _Giorgos Kalogeropoulos_, Aug 18 2023