[seqfan] Re: Unitary RMPN

[M. F. Hasler]

On Wed, Jan 29, 2014 at 4:16 AM, <hv at crypt.org> wrote:

> Hi, each prime factor of n must divide a(n), and each prime power dividing
> a(n) must be >= n. That implies 72 divides a(6), and it is easy to show
> 72 and 144 do not satisfy the requirement but 216 does, so a(n)=216.
>
> For n=10 we require 400 dividing a(10); working by hand, I found a better
> solution a(10) ?= 2^5 . 3^3 . 5^3 . 7^2; I don't know if that's the first,
> but it seems likely.
>
>
Indeed the first two 11/10-URMPN are : 5292000,650916000.


I don't know of an effective way to find these for the general case though,
> or to verify a value is the first possible, other than by crude exhaustive
> search.
>

(That's what I did (*shame*). Took about a minute, thinking is slower...)

NB: see also sequence A145681 <https://oeis.org/A145681> (3/2) and
certainly many others
(xrefs heavily missing) ! (4/3: see A144949 <https://oeis.org/A144949>-51)

Maximilian



> zbi74583.boat at orange.zero.jp wrote:
> :    Hi,Seqfan
> :
> :    [ Unitary Rational Multiply Perfect Number ]
> :
> :    Definition of URMPN :
> :    UnitarySigma(n)=k*n
> :              Where k is rational number
> :    Ex.
> :    k=3/2
> :    2,20,24,360,816,....
> :    k=5/3
> :    12,18,2^6*3*7*13,2^6*3^2*5*7*13,2^15*3^6*5*7*11*19*37*73*83*331
> :
> :    Definition of ((n+1)/n )URMPN :
> :    UnitarySigma(m)=(n+1)/n*m
> :             Wnere n is positive integer
> :
> :    Definition of Sequence a(n) :
> :    The smallest number which is ((n+1)/n)URMPN
> :
>  6,2,3,4,5,216,7,8,9,2^15*3^3*5^4*7^2*79*83*157*313*331,11,-,13,-,-,16,17,-,19,....
> :             Where "-" means unknown
> :
> :    If n is prime power then a(n)=n
> :
> :    Could anuone confirm a(6) and a(10) and compute more terms which are
> not
> :prime power?
> :    I am sure that a(12) has many digit
> :
> :
> :
> :    Yasutoshi
> :
> :
>