A198471 -id:A198471 - cp's OEIS frontend

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

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

Offset: 1

Joseph L. Pe, Mar 19 2008

nonn

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).

			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.

Index entries for sequences where any odd perfect numbers must occur

Cf. A005100, A198470, A198471.

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

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

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

A254278 Numbers that are smaller than the sum of their deficient divisors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 98, 99, 101, 102, 103, 104, 105

Offset: 1

Timothy L. Tiffin, Jul 15 2016

nonn

Contains every deficient number larger than 1 and every primitive abundant number (A071395).

			a(16) = 20 since 20 is smaller than 22, which is the sum of its deficient divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005100, supersequence of A071395 and A198471, A091191.

Select[Range@ 105, Function[n, n < Total@ Select[Divisors@ n, DivisorSigma[1, #] < 2 # &]]] (* Michael De Vlieger, Jul 16 2016 *)

isok(n) = n < sumdiv(n, d, d*(sigma(d) < 2*d)); \\ Michel Marcus, Jun 26 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI