greedy computation of pseudoperfect numbers?

[Maximilian Hasler]

> At 8:45 AM -0400 4/8/08, Maximilian Hasler wrote:
>  >>  and forbidding divisor one A136446, a greedy method _seems_ to always
>  (...)  (Perhaps your PARI code does this. I don't read PARI.) (...)

just to clarify, the ">>" indicate that it was not me who wrote that
(nor is it my PARI code)

Concerning the questions from Frank in the end of his reply,
did you have a look at that Benkoski & Erdös paper ?
(Math.Comp.28 (1974) 617-23)
They discuss several inequalities of that type (written as sum(1/d[i]) < ...)

They also mention an "old conjecture" for which Erdös offers $300 to settle it.
I thought about a related sequence, and I think the following is quite
interesting and not in OEIS:

2,3,7,11,29,53,107,211,431,853,1709,3433,6857,13709,27427,54851
a(n)=least prime such that all subsets of { a(1),...,a(n) } have a different sum

I will submit this in due form in some while (as id:A138000),
but if somebody has an interesting remark, I could add it directly in
the initial submission. I may send a copy of my full submission to
interested people but maybe not to the whole list.

Maximilian
PS: technical issue: personally, I like getting mails which concern me
directly (ie with my name in To: or CC:), even if a copy is sent to
seqfan:
my gmail won't show it twice, and (a) it's faster than through the
list and (b) I can see that it is addressed to me and not only to the
list.
An earlier comment today (yesterday?) about CC-ing makes me doubt if
such is the preference for everybody. If not, please tell me!




At 8:45 AM -0400 4/8/08, Maximilian Hasler wrote:
>> For both sorts of pseudoperfect numbers, allowing divisor one A005835,
>>  and forbidding divisor one A136446, a greedy method _seems_ to always
>>  work for deciding whther a given n in in the sequence:

Let d be the list of divisors of n (we can include or exclude 1).  Compute
terms of the polynomial product_i (1+x^d[i]) up to the x^n term.  (Perhaps
your PARI code does this. I don't read PARI.) If the x^n term is nonzero,
then n is the sum of a subset of the divisors in d.  This is fairly fast
and deterministic.  Using this method, the first 10 odd terms of A136446 are

945, 1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985

which starts the same as A005231, odd abundant numbers.  Here is the
Mathematica code:

t = {}; n = 1; While[Length[t] < 10, n = n + 2;

Best regards,

Tony