practical numbers

[franktaw at netscape.net]

To answer the second question first, it suffices to show that every 
number up to and including ceiling(n/2) is a sum of the divisors of n.  
Except for n=3, that can be floor(n/2) instead.  (If n is odd, 2 is not 
a sum; if n is even, every m with n/2 <= m < n is a sum n/2 plus a 
number less than n/2.)  This answers the first question, essentially in 
the negative: 3 is the only such number.

More interesting, perhaps: what about numbers such that every number up 
to n, with only 1 exception, are the sum of divisors of n?  (Note that 
this has to be n in this case, not sigma(n), since the possible sums 
are symmetric with respect to sigma(n) - if a set S of divisors sums to 
m, then the complement of S, with respect to the set of divisors of n, 
sums to sigma(n) - m.)

70 is one such number, skipping only 4; I think 70p for any prime 5 <= 
p < 70 does the same.  945 is another, skipping 2.  (945 is the 
smallest odd abundant number.)  I think that these values (70, 350, 
490, 770, 910, 945) are the only ones up to 1000.

The number skipped is always even; in fact, it always one less than a 
prime divisor p of n (and 1 more than the sum of the divisors of a 
practical divisor of n - which implies that that practical divisor is a 
square or twice a square).  Except for n=3, there must be another prime 
divisor q of n with p < q < 2p.

Franklin T. Adams-Watters


-----Original Message-----
From: davidwwilson at comcast.net

Assuming Franklin's observation to be correct, why not 
 
%C A005153 Also, n such that all k <= n are sums of distinct divisors 
of n. 
 
Some poorly-thought-out questions before I head off to work: 
 
Are there not-quite-practical numbers, that is, numbers such that n-1 
is the only number < n which is not the sum of distinct divisors of n? 
 
If not, is there some interesting bound f(n) < n such that if all k <= 
f(n) are sums of distinct divisors of n, then n is also a sum of 
distinct divisors of n? 
 
----- Original Message ----- From: <franktaw at netscape.net> 
To: <tanyakh at TanyaKhovanova.com>; <seqfan at ext.jussieu.fr> 
Sent: Monday, November 27, 2006 12:04 PM 
Subject: Re: practical numbers 
 
> The definitions are equivalent. The condition cited in the Wikipedia 
 > article, to make everything up to n a sum of the divisors of n, is > 
sufficient to make everything up to sigma(n) be such a sum. 
> 
> Franklin T. Adams-Watters 
 


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