ceiling2n13

ceiling2n13 is an informal term used in some mathematical discussions to denote the function f(n) = ceil(2n/13) for integers n ≥ 0. It is not a standard notation in formal mathematics, but it serves as a compact shorthand in problems that involve rounding up the value of 2n/13.

Definition and notation: For each integer n ≥ 0, f(n) is the smallest integer m such that m ≥ 2n/13. Equivalently, f(n) = ceil(2n/13). The function is a nondecreasing step function, increasing only when n passes thresholds where 2n/13 crosses an integer.

Basic properties: f(0) = 0, and as n grows, f(n) remains constant over short intervals and then jumps by 1 at certain n. Because 2/13 is a positive rational with denominator 13, the threshold points are spaced roughly every 6 or 7 steps in n, reflecting the ratio 2/13. This makes f(n) useful in problems that require distributing a quantity into 13-sized or related groups while guaranteeing a whole number of containers or slots.

Examples:

- n = 0 gives f(n) = 0.

- n = 1 to 6 gives f(n) = 1.

- n = 7 to 13 gives f(n) = 2.

- n = 14 to 19 gives f(n) = 3.

These illustrate the upward steps of the function.

Applications: The concept appears in discrete optimization, rounding in algorithm design, and resource allocation scenarios where quantities proportional to n must be rounded up after division by 13. It also relates to the general ceil function and to rounding identities such as ceil(x) = -floor(-x).

See also: Ceil function, Floor function, Ceiling division, Rounding functions.