# "A dream" of a series :-)

Gottfried Helms Annette.Warlich at t-online.de
Wed Jun 4 15:56:35 CEST 2008

```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,

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

```