A124184: prime factorization of 1?

[zak seidov]

OK, 
if as a result of this thread,
the sequence "A124184 without first 1" 
is worth submitting, 
I suggest that Frank'll do it best 
(with or without me), 
otherwise the mathematics is well over my head and 
I couldn't do it properly.

Thanks all for the pure "math" discn,
WADR, Zak 

--- Jonathan Post <jvospost3 at gmail.com> wrote:

> Excellent comments, Frank!  I hope these are added
> by njas to Zak's (in my
> opinion) nice sequence.
> 
> -- Jonathan Vos Post
> 
> On 12/7/06, franktaw at netscape.net
> <franktaw at netscape.net> wrote:
> >
> > I can confirm Zak's values.  (Except that 1 really
> does belong in the
> > sequence.)
> >
> > We really also should have the sequence of the
> numbers that are not
> > divisible by at least one of the (positive)
> exponents in their prime
> > factorization:
> >
> >
>
8,9,25,32,40,45,49,56,63,64,75,81,88,96,99,104,117,121,125,128,136,147,
> >
>
152,153,162,169,171,175,184,200,207,224,225,232,243,245,248,250,261,275,
> >
>
279,280,288,289,296,315,320,325,328,333,343,344,352,361,363,369,376,384,
> >
>
387,392,405,416,423,424,425,440,441,448,472,475,477,486,488,495,500
> >
> > In this case, 1 does not belong; it has no prime
> exponents, so there
> > aren't any of them that fail to divide it.
> >
> > Also, in this case we can't get around it by
> talking about 0 exponents;
> > if we included these, the sequence would be the
> natural numbers.
> >
> > Except for 1, every member of A124184 is in this
> sequence.  Also, this
> > sequence does not contain any square-free numbers.
> >
> > Franklin T. Adams-Watters
> >
> >
> > -----Original Message-----
> > From: zakseidov at yahoo.com
> >
> > Dear seqfans,
> >
> > In A124184,
> > shouldn't we omit 1
> > (I omit it, while Leroy includes)?
> >
> > What is prime factorization of 1?
> >
> > Mmca gives:
> >
> > FactorInteger[1]
> > {}
> >
> > Wiki says:
> > 1 has no prime factors. (1 is the empty product)
> >
> > Also, is there any special name of numbers with
> > all  prime-factorization exponents >1 ?
> >
> > In A124184, we have only such numbers!
> >
> > Thanks, Zak
> >
> >
> > (***************************************)
> > %I A124184
> > %S A124184
> >
>
8,9,25,32,49,64,81,121,125,128,169,225,243,289,343,361,441,512,529,625,72
> >
>
9,841,961,1000,1024,1089,1225,1331,1369,1521,1681,1849,2025,2048,2187,219
> >
>
7,2209,2401,2601,2744,2809,3025,3249,3481,3721,3969,4096,4225,4489,4761,4
> >
>
913,5041,5329,5625,5929,6125,6241,6561,6859,6889,7225,7569,7776,7921,8000
> > .
> > %K A124184 nonn,new
> > %O A124184 1,2
> > %D A124184 More terms from Zak Seidov
> > (zakseidov(AT)yahoo.com), Dec 08 2006
> >
> >
> >
> >
> >
>
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> 



 
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