A386316 a(n) = the minimum value of (x + 2)*(y + 2) such that x*y >= n.

4, 9, 12, 15, 16, 20, 20, 24, 24, 25, 28, 30, 30, 35, 35, 35, 36, 40, 40, 42, 42, 45, 48, 48, 48, 49, 54, 54, 54, 56, 56, 60, 60, 63, 63, 63, 64, 70, 70, 70, 70, 72, 72, 77, 77, 77, 80, 80, 80, 81, 84, 88, 88, 88, 88, 90, 90, 96, 96, 96, 96, 99, 99, 99, 100, 104, 104, 108

Offset: 0

Author

Ben Orlin, Jul 18 2025

Keywords

• nonn
• easy

Comments

Smallest number of elements in a rectangular array with at least n interior elements.

If baking square brownies in a rectangular pan, a(n) is the minimum number of brownies required to have at least n gooey center brownies.

a(n) = the minimum of A386318(m) for m >= n. If n < k^2, then the value of m that achieves this minimum is also strictly less than k^2.

Conjecture: As n grows, the length of the longest run grows without bound. The first run of length 50 begins at a(506269) = 509120. Up to 10^6 terms, the longest runs are of length 59.

Examples

			a(5) = a(6) = 20 because the 4 X 5 array is the smallest with at least 5 interior elements, and the smallest with at least 6 interior elements.
		

Crossrefs

Cf. A386318.

Programs

Mathematica

Table[Minimize[{(x+2)(y+2), x y >= n && x>=0 && y>=0}, {x,y}, Integers][[1]], {n, 0, 67}] (* Giovanni Resta, Jul 21 2025 *)

PARI

a(n) = my(m=oo, mm); for (x=0, n, for (y=0, n, if ((x*y >= n) && (mm=(x + 2)*(y + 2)) <= m, m = mm););); m; \\ Michel Marcus, Jul 21 2025

Python

import math
def a(n):
    if n == 0: return 4
    min_val = 3*(n+2)
    for x in range(1, math.isqrt(n)+1):
        y = (n + x - 1) // x
        if (x + 2) * (y + 2) < min_val:
                min_val = (x + 2) * (y + 2)
    return min_val

Formula

a(k^2) = (k+2)^2 = A386318(k^2).

a(k^2 - 1) = (k+1)(k+3) = A386318(k^2 - 1).