Self Power Numbers - Unfinished business!

[Alexander Povolotsky]

In addition to "self-power" numbers one might also consider "self-factorial"
numbers ;-).
For example 24

where 2=2!
         24=4!

On Feb 2, 2008 10:01 AM, N. J. A. Sloane <njas at research.att.com> wrote:

>
> Dear Seqfans,  (The OEIS machine is still down)
>
> Back in 2005 there was a discussion which began like this (I've edited
> it a bit):
>
> From: "Eric Angelini" <keynews.tv at skynet.be>
> Date: Tue, 5 Jul 2005 15:50:24 +0200
> Subject: Self-powers numbers (SPN)
>
> The number N=325648 might be called a self-power
> number because:
>
> "3" means "a cube is visible in N"
>    (yes, it is "8" -- 8=2*2*2)
> "2" means "a square is visible in N"
>    (yes, it is "25" -- 25=5*5)
>    (4 is ok too, being 2*2)
> "5" means "a power 5 is visible in N"
>    (yes, it is "32" = 2*2*2*2*2)
> "6" means "a power 6 is visible in N"
>    (yes, it is "64" = 2*2*2*2*2*2)
> "4" means "a power 4 is visible in N"
>    (yes, it is "256" = 4*4*4*4*)
> "8" means "a power 8 is visible in N"
>    (yes, it is "256" =2*2*2*2*2*2*2*2)
>
> N=832564 is a SPN too, of course.
>
> ["visible" means "as a whole": "25" is
> NOT visible in 235]
>
> Question:
> Can someone compute all such SPN _which
> don't include any 0's or 1's_ ?
>
> This restriction applies because 0^a=0
> and 1^b=1, which brings a lot of unwan-
> ted SPN like 117 or 308:
>
> "1" means "a power 1 is visible in N"
>    (yes, it is "1" -- 1^1=1)
> "7" means "a power 7 is visible in N"
>    (yes, it is "1" again -- 1^7=1)
>
> or
>
> "3" means "a cube is visible in N"
>    (yes, it is "0" -- 0*0*0=0)
> "0" means "a power 0 is visible in N"
>    (yes, it is "0" again -- any a^0=0)
> "8" means "a power 8 is visible in N"
>    (yes, it is "0" again and again
>                              -- 0^8=0)
> Best,
> Eric
>
> _______________________________________________
>
> Date: Tue, 05 Jul 2005 10:37:47 -0400
> From: Jack Brennen <jb at brennen.net>
>
> Eric Angelini wrote:
> > Question:
> > Can someone compute all such SPN _which
> > don't include any 0's or 1's_ ?
>
> There are an infinite number of course...
>
> For example, any sequence of digits 2-9 which contains this
> string:
>
>   2799363256438443359375
>
> Showing this is an SPN is left as an exercise for the reader.
>
> -------------------------------------
>
> >From njas at research.att.com  Wed Jul  6 15:47:33 2005
> Date: Wed, 6 Jul 2005 15:47:30 -0400 (EDT)
> From: "N. J. A. Sloane" <njas at research.att.com>
> Subject: Re: Self-powers numbers (SPN)
>
> Dear Seqfans,   There were a bunch of messages yesterday
> about Self-powers numbers (SPN).   But it looks like
> no one has yet submitted the sequence to the OEIS!
>
> Would someone kindly work out the first few terms
> and send it in?
>
> Thanks!
>
> NJAS
>
>
> -------------------------------------------
>
> Feb 02 2008:  Would someone kindly work out the first few SPN's
> and send them to me? Better use email, the OEIS machine is down.
>
> Neil
>
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