A357714 a(n) is the number of equations in the set E_{n,b} := {x+2^b*y=n^b, 2^b*x+3^b*y=n^b, ..., k^b*x+(k+1)^b*y=n^b, ..., n^b*x+(n+1)^b*y=n^b} which admit at least one nonnegative integer solution when b is sufficiently large.

1, 2, 3, 4, 3, 5, 4, 6, 5, 6, 4, 8, 5, 7, 7, 8, 5, 9, 5, 9, 8, 8, 6, 12, 7, 8, 8, 10, 6, 12, 7, 11, 9, 9, 9, 14, 7, 9, 9, 13, 7, 13, 8, 12, 12, 10, 8, 16, 9, 12, 10, 12, 8, 14, 10, 14, 11, 11, 9, 19, 9, 11, 13, 14, 11, 15, 9, 13, 11, 15, 9, 19, 10, 12, 14, 14, 12, 16, 10, 18, 13

Offset: 1

Author

Luca Onnis, Oct 10 2022

Keywords

• nonn

Comments

Defining a(n,b) as the number of equations of the set E_{n,b} which admit at least one nonnegative integer solution, it's possible to prove the existence of b_0 such that for all b > b_0, a(n,b) = a(n) whose value does not depend on b anymore.

a(n) is the number of positive integers k such that k(k+1) <= n or k divides n or k+1 divides n.

Examples

			a(11) = 4 since for all b >= 29 the number of equations of the set E_{11,b} which admit at least one nonnegative integer solution is exactly equal to 4.
a(4) = 4 since for all b >= 1 the number of equations of the set E_{11,b} which admit at least one nonnegative integer solution is exactly equal to 4.
		

Crossrefs

Cf. A000005, A356770.

Programs

Mathematica

Table[Ceiling[Sqrt[n] - 3/2] + Length[Divisors[n]], {n, 1, 100}]

Formula

a(n) = ceiling(sqrt(n) - 3/2) + A000005(n).

a(n) ~ A356770(n)/2 as n->infinity.

a(n) <= A356770(n) for all n >= 1.