A187795 Sum of the abundant divisors of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 20, 0, 0, 0, 36, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 66, 0, 0, 0, 60, 0, 42, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0, 0, 72, 0, 56, 0, 0, 0, 122, 0, 0, 0, 0, 0, 66, 0, 0, 0, 70, 0, 162, 0, 0, 0, 0, 0, 78, 0, 140, 0, 0, 0, 138, 0, 0, 0, 88, 0, 138, 0, 0, 0, 0, 0, 180

Offset: 1

Author

Timothy L. Tiffin, Jan 06 2013

Keywords

• nonn
• easy

Comments

Sum of divisors d of n with sigma(d) > 2*d.

a(n) = n when n is a primitive abundant number (A091191). - Alonso del Arte, Jan 19 2013

Examples

			a(12) = 12 because the divisors of 12 are 1, 2, 3, 4, 6, 12, but of those only 12 is abundant.
a(13) = 0 because the divisors of 13 are 1 and 13, neither of which is abundant.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A000203, A005101, A080224, A270660, A271113, A294889, A294930, A294937.

Programs

Maple

A187795 := proc(n)
    local a,d;
    a :=0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) > 2* d then
            a := a+d ;
        end if;
    end do:
    return a;
end proc:
seq(A187795(n),n=1..100) ; # R. J. Mathar, Apr 27 2017

Mathematica

Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] > 2 # &], {n, 96}] (* Michael De Vlieger, Jul 16 2016 *)

PARI

a(n)=sumdiv(n,d,(sigma(d,-1)>2)*d) \\ Charles R Greathouse IV, Jan 15 2013

Python

from sympy import divisors, divisor_sigma
def A187795(n): return sum(d for d in divisors(n,generator=True) if divisor_sigma(d) > 2*d) # Chai Wah Wu, Sep 22 2021

Formula

From Antti Karttunen, Nov 14 2017: (Start)

a(n) = Sum_{d|n} A294937(d)*d.

a(n) = A294889(n) + (A294937(n)*n).

If A294889(n) > 0, then a(n) = A294889(n)+n, otherwise a(n) = A294930(n)*n.

a(n) + A187794(n) + A187793(n) = A000203(n).

(End)