Strange Numbers

[Turtle]

Originally Posted by Bombadil

I was just looking at your list of strange numbers wondering...

That's the ticket! Man after my own heart. Nothing quickens my pulse quite like looking at lists of numbers and just wondering.

Originally Posted by LaBombadilly

As for what to do next, I haven’t tried writing it out yet but I suspect that by choosing the form of a strange number as with p and q being primes such that and and solving for one of them, that a contradiction may be reached but I haven’t had the chance to try it to find out for sure so maybe its just a dead end.

Roger. Standing by. A dead end is always an opportunity to review a path.

Originally Posted by Bomba Real Deal

P.S. I see that it also can easily be used to generate strange numbers in other number sets as well so I wonder what other patterns might be found with it.

Oh yeah baby! The replaceable subtrahends {13, 57, 993...} may be generalized as where P is a Perfect Number. I was a little excited at seeing the 13 when you first put it up, as it is a Mersenne prime, however that is not the case for 57 or 993. Nonetheless we can, and must, invoke Mersenne Primes when talking about Perfect Numbers. To whit:

Originally Posted by Mathworld

Perfect numbers are also intimately connected with a class of numbers known as Mersenne primes, which are prime numbers of the form M_p = 2^p - 1. ...

Perfect Number -- from Wolfram MathWorld

Thanks again Bombadil. Very insightful. To infinity & beyond at the speed of a racing tortoise.

[Turtle]

Just thinking out loud as I puzzle over that anomlous Strange Anomaly yet properly accounted for (not to mention the unsurety of whether or not it has cohorts), namely that rogue 54. {(1, 54) (2, 27) (3, 18) (6, 9):} So it's 6*9 and I keep going back to how 9 is a Perfect Square of 3, but everything else we have found is going with powers of two, but then all even powers of 2 are Perfect Squares, so now if ......but ........... then ...............

[Turtle]

Hello Big Dog!? Hello Alexander!?? Bombadill?? Modest? Is there anybody in there??? Something must have sidetracked me here for a couple of months; I can't imagine.

Any further progress on the strange sieving software fellas?? We have not yet so well managed these Unusual sets as to preclude the possibility that we have missed some members by not forwarding an exhaustive discrete search of every possible candidate. Can't have that now can we? 'Course not silly rhetorical turtle! (Good grief & oh bother; I seem to be talking to myself again. )