Semi Integer Multiply Perfect

Thu May 1 03:43:45 CEST 2008

    Dear Richard Mather

    >     I found examples of "3/2 Multiply Unitary Perfect Number"  
    > 
    >         Sigma(m)=3/2*m

    Sorry I did a mistake.

    I should have written as follows.

       UnitarySigma(m)=3/2*m

    Yasutoshi

* Peter Pein <petsie at dordos.net> [May 01. 2008 11:27]:
> Dear seqfans,
> 
> propably I overlook something important, but I can't figure out what.
> 
> The definition of http://www.research.att.com/~njas/sequences/A073071 is:
> "Least k such that k! > prime(1)*prime(2)*...*prime(n) where prime(n) is the
> n-th prime."
> and it starts with n=1.
> 

pp(1)=2
and the defn says k! > pp
the first k s.t.  k!>2 is indeed 3

pp(2) = 6
and k! > 6 for k>=4
as given

> [...]

cheers,   jj

pp> From seqfan-owner at ext.jussieu.fr  Thu May  1 03:27:20 2008
pp> Date: Thu, 01 May 2008 03:26:58 +0200
pp> From: Peter Pein <petsie at dordos.net>
pp> To: seqfan at ext.jussieu.fr
pp> Subject: Is A073071 correct?
pp> ...
pp> propably I overlook something important, but I can't figure out what.
pp> 
pp> The definition of http://www.research.att.com/~njas/sequences/A073071 is:
pp> "Least k such that k! > prime(1)*prime(2)*...*prime(n) where prime(n) is the
pp> n-th prime."
pp> and it starts with n=1.
pp> ....

Indeed, the numbers in A073071 do not have anything in common with its
definition. One would try variants like replacing the greater sign
with greater-or-equal, shifting indices, etc. These cosmetic changes would
not make a real difference, because the first differences of A073071
as given are systematically slightly larger than the first differences
of the sequence defined by k!>A002110(n). So the way out is to replace
the numbers of A073071 as proposed by Peter,

3,4,5,6,7,8,10,11,12,14,15,16,17,19,20,21,23,24,25,27,28,29,31,32,33,34,36,37,38,40,41,42,44,45,
46,47,49,50,51,53,54,55,57,58,59,60,62,63,64,66,67,68,70,71,72,73,75,76,77,79,80,81,83,84,85,86,
88,89,90,92,93,94,96,97,98,99,101,102,103,105,106,107,108,110,111,112,114,115,116,118,119,120,121,
123,124,125,127,128,129,130,132,133,134,136,137,138,139,141,142,143,145,146,147,149,150,151,152,154,
155,156

To avoid confusion, one might add a comment to A073071:
"If the sign in the definition is replaced by larger-or-equal, we get A048964."

Those with a more entrepreneurial spirit could think about the sequence
"Least k such that prime(1)*prime(2)*...*prime(k)>n!" which seems not to
be in the OEIS.

Richard

pp> From seqfan-owner at ext.jussieu.fr  Thu May  1 03:27:20 2008
pp> Date: Thu, 01 May 2008 03:26:58 +0200
pp> From: Peter Pein <petsie at dordos.net>
pp> To: seqfan at ext.jussieu.fr
pp> Subject: Is A073071 correct?
pp> ...
pp> propably I overlook something important, but I can't figure out what.
pp> 
pp> The definition of http://www.research.att.com/~njas/sequences/A073071 is:
pp> "Least k such that k! > prime(1)*prime(2)*...*prime(n) where prime(n) is the
pp> n-th prime."
pp> and it starts with n=1.
pp> ....

Another possibility is that the sequence originally said:

Least k such that k! > P(1)*P(2)*...*P(n).

and I guessed that P(i) meant prime(i).  But maybe it meant
partitions(i) ?  Benoit, do you remember what
the original definition was for A073071?

In any case, I will correct the present version.

Neil

pp> From seqfan-owner at ext.jussieu.fr  Thu May  1 03:27:20 2008
pp> Date: Thu, 01 May 2008 03:26:58 +0200
pp> From: Peter Pein <petsie at dordos.net>
pp> To: seqfan at ext.jussieu.fr
pp> Subject: Is A073071 correct?
pp> ...
pp> propably I overlook something important, but I can't figure out what.
pp> 
pp> The definition of http://www.research.att.com/~njas/sequences/A073071 is:
pp> "Least k such that k! > prime(1)*prime(2)*...*prime(n) where prime(n) is the
pp> n-th prime."
pp> and it starts with n=1.
pp> ....

njas> From seqfan-owner at ext.jussieu.fr  Thu May  1 15:29:27 2008
njas> Date: Thu, 1 May 2008 09:29:03 -0400
njas> From: "N. J. A. Sloane" <njas at research.att.com>
njas> To: mathar at strw.leidenuniv.nl, seqfan at ext.jussieu.fr
njas> Subject: Re: Is A073071 correct?
njas> Cc: benoit7848c at orange.fr, njas at research.att.com
njas> Another possibility is that the sequence originally said:
njas> 
njas> Least k such that k! > P(1)*P(2)*...*P(n).
njas> 
njas> and I guessed that P(i) meant prime(i).  But maybe it meant
njas> partitions(i) ?  Benoit, do you remember what
njas>...

The interpretation P=partitions keeps the growth rate close to what
we are searching for, but it does not (yet) fit the original
A073071 version, I think. The two test cases are:

"Smallest k such that k! > A058694(n)", n>=1 :
2,3,4,5,6,7,8,10,11,13,14,16,18,19,21,23,25,26,28,30,32,34,36,38,40,42,44,47,49,51,53,55,58,60,
62,65,67,70,72,74,77,79,82,84,87,90,92,95,97,100,103,105,108,111,114,116,119,122,125,128,131,134,
136,139,142,145,148,151,154,157,160,163,166,169,172,176,179,182,185,188

"Smallest k such that k! >= A058694(n)", n>=1 , differs in first 3 terms:
1,2,3,5,6,7,8,10,11,13,14,16,18,19,21,23,25,26,28,30,32,34,36,38,40,42,44,47,49,51,53,55,58,60,
62,65,67,70,72,74,77,79,82,84,87,90,92,95,97,100,103,105,108,111,114,116,119,122,125,128,131,134,
136,139,142,145,148,151,154,157,160,163,166,169,172,176,179,182,185,188

A000041 := proc(n)
end:
A058694 := proc(n)
end:

A000001 := proc(n)
end:

Yes.