A099475 Number of divisors d of n such that d+2 is also a divisor of n.

0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 3, 0, 0, 2, 1, 0, 1, 0, 1, 1, 0, 0, 4, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 3, 0, 0, 1, 2, 0, 1, 0, 1, 2, 0, 0, 4, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 5, 0, 0, 2, 1, 0, 1, 0, 1, 1, 1, 0, 4, 0, 0, 2, 1, 0, 1, 0, 2, 1, 0, 0, 4, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 4, 0, 0, 2, 1, 0, 1, 0, 1, 3

Offset: 1

Author

Reinhard Zumkeller, Oct 18 2004

Keywords

• nonn
• easy

Comments

Number of r X s rectangles with integer sides such that r < s, r + s = 2n, r | s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 24 2020

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007862 (similar but with d+1 instead).

Cf. A008585, A008586, A008588.

Cf. A059267, A099476, A099477.

Programs

Maple

A099475:= proc(n)
local d;
  d:= numtheory:-divisors(n);
nops(d intersect map(`+`,d,2))
end proc:
map(A099475,[$1..1000]); # Robert Israel, Jun 19 2015

Mathematica

a[n_] := DivisorSum[n, Boole[Divisible[n, #+2]]&]; Array[a, 105] (* Jean-François Alcover, Dec 07 2015 *)

PARI

A099475(n) = { sumdiv(n, d, ! (n % (d+2))) } \\ Michel Marcus, Jun 18 2015

Formula

0 <= a(n) <= a(m*n) for all m>0;

a(A099477(n)) = 0; a(A059267(n)) > 0;

a(A099476(n)) = n and a(m) <> n for m < A099476(n).

For n>0: a(A008585(n))>0, a(A008586(n))>0 and a(A008588(n))>0.

a(n) = Sum_{i=1..n-1} chi((2*n-i)/i) * chi(i*(2*n-i)/(2*n-2*i)), where chi(n) = 1 - ceiling(n) + floor(n). - Wesley Ivan Hurt, Apr 24 2020