[seqfan] Near-perfect numbers

Mon Nov 1 19:06:41 CET 2010

```Dear SeqFans,

I have just submitted the following sequence:

%I A181595
%S A181595 12,18,20,24,40,56,88,104,196,224,234,368,464,650,992,1504,1888,1952,
%T A181595 3724,5624,9112,11096,13736,15376,15872,16256,17816,24448,28544,30592,
%U A181595 32128,77744,98048,122624,128768,130304,174592,396896,507392
%N A181595 Near-perfect numbers, i.e. abundant numbers m for which there exists a proper divisor d=d(m) such that sigma(m)-d=2m.
%C A181595 Union of this sequence and A005820 is A153501.
%e A181595 12 is near-perfect with d(12)=4. Indeed, the sum of other proper divisors is 1+2+3+6=12.
%Y A181595 A000396 A005101 A153501 A005820
%K A181595 nonn
%O A181595 1,1

The corresponding sequence of suitable {d} is

%I A181596
%S A181596 4,3,2,12,10,8,4,2,7,4,3,8,2,2,32,16,4,2,532,152,136,8,68,31,992,128,8,
%T A181596 64,32,16,4,8,128,32,8,2,43648,2528,32
%N A181596 a(n)=sigma(A181595(n))-2*A181595(n)
%C A181596 a(n) is a proper divisor of A181595(n).
%e A181596 Since A181595(1)=12, then a(1)=sigma(12)-2*12=28-24=4.
%Y A181596 A181595 A000396 A005101 A153501 A005820
%K A181596 nonn
%O A181596 1,1

One can notice that this sequence contains many powers of 2. E.g., 8 corresponds to the following
terms of A181595:

56,368,11096,17816,77744,128768,2087936,2291936,13174976,35021696,
45335936,381236216,4856970752,6800228816,8589344768,...
It is natural to call these numbers (N\{8})-perfect ( if m is  (N\{8})-perfect, then 8|m and
sigma(m)-8=2m). This sequence is not in OEIS.
Note that (N\{2})-perfect numbers for which sigma(m)-2=2m are all even numbers of A045768.
On the other hand, to a term k of A181596 which is NOT a power of 2, it seems, corresponds a finite or very fast growing sequence of  (N\{k})-perfect numbers. It is interesting to look at some of such sequences.

Regards,