"A dream" of a series :-)

[Gottfried Helms]

A dream of a series...

Consider the exponentialseries

   E0 =  1 + x/1! + x^2/2! + x^3/3!  + ...

Because I considered the following problem in a roughly
similar way,

 see http://groups.google.com/group/alt.math.recreational/browse_thread/thread/606d98fef53db8ff/7aa1d8f1f840736e?lnk=st&q=Gottfried+Helms+infinite+product+exp#7aa1d8f1f840736e

a correspondent asked me, whether I knew the following about
a factoring of the exponential-series.

The idea due to owen james maresh ---------------------------------

Define E1 by extracting the factor f1=(1+1*x):

   E1 = E0 /(1+1*x) = 1 + 1/2*x^2 - 1/3*x^3 + 3/8*x^4 - 11/30*x^5 + 53/144*x^6

Then define E2 by extracting the factor f2=(1+1/2*x^2):

   E2 = 1 - 1/3*x^3 + 3/8*x^4 - 1/5*x^5 + 13/72*x^6

Proceed this way to any arbitrary degree.

-----

Formally we get then (not discussing convergence)

 E0 = f1 * f2 * f3 * f4 * ....
    = (1+x)(1+1/2x^2)(1-1/3x^3) ...

and the list of factors f1,f2,f3, begins then

                                  x+1 = f1
                            1/2*x^2+1 = f2
                           -1/3*x^3+1 = f3
                            3/8*x^4+1 = f4
                           -1/5*x^5+1 = f5
                          13/72*x^6+1
                           -1/7*x^7+1
                         27/128*x^8+1
                          -8/81*x^9+1
                        91/800*x^10+1
                         -1/11*x^11+1
                    1213/13824*x^12+1
                         -1/13*x^13+1
                      505/6272*x^14+1
                   -1919/30375*x^15+1
                    2955/32768*x^16+1
                         -1/17*x^17+1
                  24557/419904*x^18+1
                         -1/19*x^19+1
              1136313/20480000*x^20+1
                 -34943/750141*x^21+1
                  12277/247808*x^22+1
                         -1/23*x^23+1
           65978519/1528823808*x^24+1
                    -624/15625*x^25+1
                 57331/1384448*x^26+1
                -58528/1594323*x^27+1
          195948483/5035261952*x^28+1
                         -1/29*x^29+1
  1052424027703/30233088000000*x^30+1

The observation, which this correspondent stumbled across, was,
that apparently at all prime indexes k the cofactor c_k has
prime denominator, which alone is already interesting.

-------------------------------------------------------------------

But there's some more in it, as I just found with a short
analysis (looking at it up to index n=64)

Consider indexes composite of primes to the first power:
   index       denominator
  6 = 2*3       2^3 * 3^2
 10 = 2*5       2^5 * 5^2
 14 = 2*7       2^7 * 7^2

index          denominator
 30 = 2*3*5     2^(3*5) * 3^(2*5) * 5^(2*3)

So we may rewrite
index          denominator
prime p = 1*p   p^1 * 1^p


For prime-powers it looks like
  4  = 2^2      2^3
  8  = 2^3      2^7
 16  = 2^4      2^15
 32  = 2^5      2^31
 64  = 2^6      2^63

  9  = 3^2      3^4
 27  = 3^3      3^13

 25  = 5^2      5^6

 49  = 7^2      7^8

and some more composites - but here I don't really have an idea:

 12  = 2^2*3    (2^3)^3 * 3^3
 24  = 2^3*3    (2^7)^3 *(3^3)^2
 48  = 2^4*3    (2^15)^3* (3^3)^5

 18  = 2 * 3^2  2^6     * (3^4)^2
 54  = 2 * 3^3  2^23    * (3^13)^2

 36  = 2^2*3^2  2^24    * 3^15

 25  = 5^2              5^6
 50  = 2*5^2    2^21 * (5^6)^2

 45  = 3^2 * 5  (3^4)^5 * 5^9
 63  = 3^2 * 7  (3^4)^7 * 7^9

Although I could not decode the composites reasonably, I feel
this series of denominators is really "a dream" for a
true seqfan... :-)

Gottfried Helms