A187793 Sum of the deficient divisors of n.

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 10, 14, 24, 24, 31, 18, 15, 20, 22, 32, 36, 24, 18, 31, 42, 40, 28, 30, 36, 32, 63, 48, 54, 48, 19, 38, 60, 56, 30, 42, 48, 44, 84, 78, 72, 48, 34, 57, 93, 72, 98, 54, 42, 72, 36, 80, 90, 60, 40, 62, 96, 104, 127, 84, 72, 68, 126, 96, 74, 72, 27

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) = sigma(n) when n is itself also deficient.

Also, a(n) agrees with the terms in A117553 except when n is a multiple (k > 1) of either a perfect number or a primitive abundant number.

Notice that a(1) = 1. The remaining fixed points are given by A125310. - Timothy L. Tiffin, Jun 23 2016

a(A028982(n)) is an odd integer. Also, if n is an odd abundant number that is not a perfect square and n has an odd number of abundant divisors (e.g., 945 has one abundant divisor and 4725 has three abundant divisors), then a(n) will also be odd: a(945) = 975 and a(4725) = 2675. - Timothy L. Tiffin, Jul 18 2016

Examples

			a(12) = 10 because the divisors of 12 are 1, 2, 3, 4, 6, 12; of these, 1, 2, 3, 4 are deficient, and they add up to 10.
		

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, A005100, A028982, A080226, A117553, A125310, A125499, A187794, A187795, A247328, A274338, A274339, A274340, A274380, A274549, A274829, A294886, A294934.

Programs

Maple

A187793 := 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:
    a ;
end proc:# R. J. Mathar, May 08 2019

Mathematica

Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] < 2 # &], {n, 72}] (* Michael De Vlieger, Jul 18 2016 *)

PARI

a(n)=sumdiv(n,d,if(sigma(d,-1)<2,d,0)) \\ Charles R Greathouse IV, Jan 07 2013

Formula

From Antti Karttunen, Nov 14 2017: (Start)

a(n) = Sum_{d|n} A294934(d)*d.

a(n) = A294886(n) + (A294934(n)*n).

a(n) + A187794(n) + A187795(n) = A000203(n).

(End)

Extensions

a(54) corrected by Charles R Greathouse IV, Jan 07 2013