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%I #29 Jun 17 2019 18:01:34
%S 6,28,90,496,8128,33550336,8589869056
%N Numbers n such that n = sum of deficient proper divisors of n.
%C Since any proper divisor of a perfect number is deficient, all perfect numbers are (trivially) included in the sequence.
%C 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?
%C 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
%C a(8) > 10^10. - _Giovanni Resta_, Jan 08 2013
%C These "deficiently perfect" numbers are pseudoperfect (A005835) and are a proper multiple of a nondeficient number (and hence abundant).
%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%e 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.
%t 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
%o (PARI) is(n)=sumdiv(n,d,(sigma(d,-1)<2 && d<n)*d)==n \\ _Charles R Greathouse IV_, Jan 17 2013
%Y Cf. A005100, A198470, A198471.
%Y Subsequence of A005835. Fixed points of A294886. Cf. also A294900.
%K nonn
%O 1,1
%A _Joseph L. Pe_, Mar 19 2008
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