OFFSET
1,1
COMMENTS
Since any proper divisor of a perfect number is deficient, all perfect numbers are (trivially) included in the sequence.
Hence the interesting terms of the sequence are its non-perfect terms, which I call "deficiently perfect". 90 is the only such term < 10^8. Are there any more?
If a(n) were defined to be those numbers that are equal to the sum of their deficient divisors, then the sequence would begin with 1. So, up to 10^10, the only non-perfect numbers in that sequence would be 1 (a deficient number) and 90 (an abundant number). - Timothy L. Tiffin, Jan 08 2013
a(8) > 10^10. - Giovanni Resta, Jan 08 2013
These "deficiently perfect" numbers are pseudoperfect (A005835) and are a proper multiple of a nondeficient number (and hence abundant).
LINKS
EXAMPLE
90 has deficient proper divisors 1, 2, 3, 5, 9, 10, 15, 45, which sum to 90. Hence 90 is a term of the sequence.
MATHEMATICA
sigdef[n_] := Module[{d, l, ct, i}, d = Drop[Divisors[n], -1]; l = Length[d]; ct = 0; For[i = 1, i <= l, i++, If[DivisorSigma[1, d[[i]]] < 2 d[[i]], ct = ct + d[[i]]]]; ct]; l = {}; For[i = 1, i <= 10^8, i++, If[sigdef[i] == i, l = Append[l, i]]]; l
PROG
(PARI) is(n)=sumdiv(n, d, (sigma(d, -1)<2 && d<n)*d)==n \\ Charles R Greathouse IV, Jan 17 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Joseph L. Pe, Mar 19 2008
STATUS
approved