[seqfan] Re: Unitary RMPN

[Jack Brennen]

a(14) = 4991499040640000
a(15) = 165375
a(18) = 235270656
a(20) = 101867327360000
a(24) = 552063590295800832

Each of a(21), a(22), a(26), a(28), a(30) > 10^22.




On 1/29/2014 2:42 PM, hv at crypt.org wrote:
> Using a crude exhaustive search I can confirm:
> a(6) = 216 [ 2^3 * 3^3 ]
> a(10) = 5,292,000 [ 2^5 * 3^3 * 5^3 * 7^2 ]
> a(12) = 10,584,000 [ 2^6 * 3^3 * 5^3 * 7^2 ]
> a(14) > 784,000,000
>
> Not sure how to make any further progres; can anyone prove even that
> a(n) exists for all n?
>
> Hugo
>
> Earlier I 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.
> :
> :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.
> :
> :Hugo
> :
> :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
> ::
> ::
> ::
> ::
> ::
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