A385454 Difference of the largest and smallest semiperimeters of an integral rectangle with area n.

0, 0, 0, 1, 0, 2, 0, 3, 4, 4, 0, 6, 0, 6, 8, 9, 0, 10, 0, 12, 12, 10, 0, 15, 16, 12, 16, 18, 0, 20, 0, 21, 20, 16, 24, 25, 0, 18, 24, 28, 0, 30, 0, 30, 32, 22, 0, 35, 36, 36, 32, 36, 0, 40, 40, 42, 36, 28, 0, 45, 0, 30, 48, 49, 48, 50, 0, 48, 44, 54, 0, 56, 0

Offset: 1

Author

James C. McMahon, Jun 29 2025

Keywords

• nonn

Comments

For all noncomposite n, a(n) = 0.

For each square k^2, a(k^2) = (k^2 + 1) - 2*k = (k-1)^2.

Examples

			The largest semiperimeter of an integral rectangle with area 9 is 10 (1 x 9 rectangle); the smallest semiperimeter is 6 (3 x 3 rectangle). The difference, a(9) = 4.
		

Links

• James C. McMahon, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A063655.

Programs

Mathematica

a[n_]:=1+n-2Median[Divisors[n]];Array[a,73]

Python

from sympy import divisors
def A385454(n):
    l = len(d:=divisors(n))
    return n+1-d[l-1>>1]-d[l>>1] # Chai Wah Wu, Jul 01 2025

Formula

a(n) = 1 + n - A063655(n).