# [seqfan] Re: Near-perfect numbers

Wed Nov 3 22:15:01 CET 2010

```Conjecture. For every k>=1, there exist infinitely many near-perfect numbers m for which (2^k)|m and sigma(m)-2^k=2*m.
This conjecture is based on the following proposition.

Proposition. If , for given k>=1, there exist infinitely many primes of the form 2^t-2^k-1, then there exist infinitely many near-perfect numbers m for which (2^k)|m and sigma(m)-2^k=2*m.
Proof. Show that M=2^(t-1)*P, where P is a prime of the form 2^t-2^k-1, is a near-perfect number  for which (2^k)|M and sigma(M)-2^k=2*M. Indeed, since, evidently, k<=t-1,
then  (2^k)|M. Besides, sigma(M)=(2^t-1)*(P+1)= (2^t-1)*(2^t-2^k) and the proposition follows
from a simple identity: sigma(M)-2*M=(2^t-1)*(2^t-2^k) -2^t*(2^t-2^k-1)=2^k.

Remark 1. In contrast of the perfect numbers, generally speaking, there exist near-perfect numbers with suitable divisor d=2^k of another forms.
Remark 2. In A181596 I corrected a(10) and a(11): a(10)=56, a(11)=78.
Remark 3. More exactly to define (N\{r})-perfect numbers as numbers m for which sigma(m)-r=2m, if r|m, otherwise sigma(m)=2m.
For example, sequence (N\{32})-perfect numbers  is union of {6,28,496}, near-perfect numbers m  for which d(m)=32, and all odd perfect numbers (if they exist).

Regards,

----- Original Message -----
From: Vladimir Shevelev <shevelev at bgu.ac.il>
Date: Monday, November 1, 2010 23:39
Subject: [seqfan]  Near-perfect numbers
To: seqfan at list.seqfan.eu

> 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,
>