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

1,2

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

Links

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

Index entries for sequences related to sums of divisors

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)

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A000203, A005100, A028982, A080226, A117553, A125310, A125499, A187794, A187795, A247328, A274338, A274339, A274340, A274380, A274549, A274829, A294886, A294934.

Sequence in context: A367502 A023888 A222085 * A284326 A117553 A331694

Adjacent sequences: A187790 A187791 A187792 * A187794 A187795 A187796

Keyword

nonn,easy

Author

Timothy L. Tiffin, Jan 06 2013

Extensions

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

Status

approved