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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.
0
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.
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
Sequence in context: A335943 A287821 A341830 * A299757 A159630 A305747
KEYWORD
nonn
AUTHOR
Luca Onnis, Oct 10 2022
STATUS
approved