A125310 Numbers n such that n = sum of deficient proper divisors of n.

6, 28, 90, 496, 8128, 33550336, 8589869056

Offset: 1

Author

Joseph L. Pe, Mar 19 2008

Keywords

• nonn

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).

Examples

			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.
		

Links

• Index entries for sequences where any odd perfect numbers must occur

Crossrefs

Cf. A005100, A198470, A198471.

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

Programs

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

PARI

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