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A187793
Sum of the deficient divisors of n.
17
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
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
KEYWORD
nonn,easy
AUTHOR
Timothy L. Tiffin, Jan 06 2013
EXTENSIONS
a(54) corrected by Charles R Greathouse IV, Jan 07 2013
STATUS
approved