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

%I A385454 #20 Jul 01 2025 01:06:05
%S A385454 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,
%T A385454 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,
%U A385454 40,42,36,28,0,45,0,30,48,49,48,50,0,48,44,54,0,56,0
%N A385454 Difference of the largest and smallest semiperimeters of an integral rectangle with area n.
%C A385454 For all noncomposite n, a(n) = 0.
%C A385454 For each square k^2, a(k^2) = (k^2 + 1) - 2*k = (k-1)^2.
%H A385454 James C. McMahon, <a href="/A385454/b385454.txt">Table of n, a(n) for n = 1..10000</a>
%F A385454 a(n) = 1 + n - A063655(n).
%e A385454 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.
%t A385454 a[n_]:=1+n-2Median[Divisors[n]];Array[a,73]
%o A385454 (Python)
%o A385454 from sympy import divisors
%o A385454 def A385454(n):
%o A385454     l = len(d:=divisors(n))
%o A385454     return n+1-d[l-1>>1]-d[l>>1] # _Chai Wah Wu_, Jul 01 2025
%Y A385454 Cf. A063655.
%K A385454 nonn
%O A385454 1,6
%A A385454 _James C. McMahon_, Jun 29 2025