A259124 If n is representable as x*y+x+y, with x>=y>1, then a(n) is the sum of all x's and y's in all such representations. Otherwise a(n)=0.

0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 5, 0, 0, 6, 6, 0, 7, 0, 7, 8, 0, 0, 17, 8, 0, 10, 9, 0, 20, 0, 10, 12, 0, 10, 34, 0, 0, 14, 23, 0, 26, 0, 13, 28, 0, 0, 43, 12, 13, 18, 15, 0, 32, 14, 29, 20, 0, 0, 67, 0, 0, 36, 32, 16, 38, 0, 19, 24, 32, 0, 76, 0, 0, 44, 21, 16, 44, 0, 57, 44

Offset: 1

Author

Alex Ratushnyak, Jun 18 2015

Keywords

• nonn

Comments

The sequence of numbers that never appear in a(n) begins: 1, 2, 3, 11, 27, 35, 51, 53, 79, 83, 89, 93, 117, 123, 125, 135, 143, 145.

The indices n at which a(n)=0 are in A254636. - Vincenzo Librandi, Jul 16 2015

Examples

			11 = 3*2 + 3 + 2, so a(11)=5.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A254636, A255361.

Programs

Maple

f:= proc(n) local D,d;
      D:= select(t -> (t >= 3 and t^2 <= n+1), numtheory:-divisors(n+1));
      add(d + (n+1)/d - 2, d = D);
end proc:
map(f, [$1..100]); # Robert Israel, Aug 05 2015

Mathematica

a[n_] := Sum[Boole[3 <= d <= Sqrt[n+1]] (d+(n+1)/d-2), {d, Divisors[n+1]}];
Array[a, 100] (* Jean-François Alcover, Jun 08 2020, after Maple *)

PARI

a(n)=sum(y=2,sqrtint(n+1)-1, my(x=(n-y)/(y+1)); if(denominator(x)==1, x+y)) \\ Charles R Greathouse IV, Jun 29 2015

Python

TOP = 100
a = [0]*TOP
for y in range(2, TOP//2):
  for x in range(y, TOP//2):
    n = x*y + x + y
    if n>=TOP: break
    a[n] += x+y
print(a[1:])

Python

from sympy import divisors
def A259124(n):
    m, c = n+1, 0
    for d in divisors(m):
        if d**2>m:
            break
        if d>=3:
            c += d+m//d-2
    return c # Chai Wah Wu, Oct 15 2024

Formula

a(n) = Sum({d: d | n+1 and 3 <= d <= sqrt(n+1)}, d + (n+1)/d - 2). - Robert Israel, Aug 05 2015