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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)



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).


Table of n, a(n) for n=1..7.

Index entries for sequences where any odd perfect numbers must occur


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.


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


(PARI) is(n)=sumdiv(n, d, (sigma(d, -1)<2 && d<n)*d)==n \\ Charles R Greathouse IV, Jan 17 2013


Cf. A005100, A198470, A198471.

Subsequence of A005835. Fixed points of A294886. Cf. also A294900.

Sequence in context: A055711 A141255 A091321 * A138874 A172141 A172132

Adjacent sequences:  A125307 A125308 A125309 * A125311 A125312 A125313




Joseph L. Pe, Mar 19 2008



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