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

1,2

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.

Links

Table of n, a(n) for n=1..81.

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.

Example

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.

Mathematica

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

Crossrefs

Cf. A000005, A356770.

Sequence in context: A335943 A287821 A341830 * A299757 A159630 A305747

Adjacent sequences: A357711 A357712 A357713 * A357715 A357716 A357717

Keyword

nonn

Author

Luca Onnis, Oct 10 2022

Status

approved