Incorrect conjecture

[franktaw at netscape.net]

The conjecture in http://www.research.att.com/~njas/sequences/A092903 
is wrong.  This conjecture is that A092903 is not the same as A005153.  
Translated, the opposite of this conjecture is that if every number up 
to m can be expressed as the sum of distinct divisors of m, then every 
number up to sigma(m) can be.  (The latter condition is known as m is 
practical.)

Following the link from A005153 to 
http://planetmath.org/encyclopedia/PracticalNumber.html, we find the 
lemma (paraphrased for non-graphical presentation): an integer m >= 2 
with factorization Product_{i=1}^k p_i^e_i with the p_i in ascending 
order is practical if and only if p_1 = 2 and, for 1 < i <= k, p_i <= 
sigma(Product_{j < i} p_j^e_j) + 1.

Proof: Let m =  Product_{i=1}^k p_i^e_i  be a number not of this form.  
If p_1 != 2, 2 cannot be represented as a sum.  If p_i > 
sigma(Product_{j<i} p_j^e_j) + 1, then sigma(Product_{j<i} p_j^e_j) + 1 
cannot be represented as the sum, and since this number is less than 
p_i, it is less than m.  Hence not every number up to m can be 
expressed as the sum of distinct divisors of m.   Thus if every number 
up to m is expressible as the sum of distinct divisors, so is every 
number up to sigma(m).  The converse is trivial, so the conditions are 
equivalent.

Hence A092903 is a duplicate of A005153.

Franklin T. Adams-Watters

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