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 A125310 Numbers n such that n = sum of deficient proper divisors of n. 10
 6, 28, 90, 496, 8128, 33550336, 8589869056 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified January 19 12:48 EST 2020. Contains 331049 sequences. (Running on oeis4.)