Mersenne numbers and the (divisors) property.

[Annette.Warlich at t-online.de]

Am 27.04.05 15:52 schrieb Gerald McGarvey:
> 
> Something to note about A007733:
> For 2 <= n <= 84, A000010(n) / A007733(n) results in the following
> sequence:
> 1 1 2 1 1 2 4 1 1 1 2 1 2 2 8 2 1 1 2 2 1 2 4 1 1 1 4 1 2 6 16 2 2 2 2 1
> 1 2 4 2 2 3 2 2 2 2 8 2 1 4 2 1 1 2 8 2 1 1 4 1 6 6 32 4 2 1 4 2 2 2 4 8
> 1 2 2 2 2 2 8 1 2 1 4
> and it seems reasonable to conjecture that A007733 always divides A000010 (
> 
> Euler totient function phi(n)).
> 
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000010
> -- Gerald

Hi Gerald -

 just recently I posted an article in the german newsgroup de.sci.mathematik
 touching that topic.

 The Length-function L(N,base) of periodicity of 2^k mod N (k=0..N)
 is clearly an integral fraction of the phi-function, at
 least for P elem PRIMES, since it must be, that
 L(P,2)*j = phi(P) = P-1
 because of
  2^(P-1) === 1  (mod p)
 and
  2^L(P,2) === 1 (mod p)

 The sequence A007733 is *somehow* multiplicative, and at a first
 glance it seems to be (but is only with exceptions):

(1)   L(P^b,base) = L(P,base) * P^(b-1)    // with exceptions

 and, with composite N = p1^b1 * p2^b2 * ... * pk^bk   and gcd(N,base)=1
                                                               N
(2)   L(N,base) = lcm(L(p1,base),L(p2,base),...L(pk,base) * -------------
                                                             p1*p2*...*pk
   // ***with exceptions ****

 Exceptions even with p elem PRIMES are for instance L(1093^b,2)
 L(1093^1,2) = 364
 L(1093^2,2) = 364
 L(1093^b,2) = 364 * 1093^(b-2)  // checked for some small b

 On the other hand, *without exceptions* are for the mersenne numbers

  L(2^n-1,2 ) = n

 Considering an attempt to finding A007733 in terms of multi-
 plicity it seems, that the *only* smooth way to define a
 formula for the lengthes L is decomposing  2^n - 1 into its prime
 factors and derive *from that* the lengthes A007733, so to say that
 the primefactorization of the mersenne numbers are the "independent"
 term, and the Length-function of an N  L(N,base) is then a
 "dependent" derivation of that prime-factorization (I don't know
 whether this way of thinking is really helpful).

 If there would be a general definition for L in a similar
 form as (1), (which were independent of the mersenne-factorization),
 or (which would be equivalent) for your above sequence,
 and this could indeed be expressed with an exponential form in b,
 then one could use that form to prove the catalan-conjecture
 in a very simple way - at least concerning powers of primes.

Gottfried Helms

reply to: helms(at)uni-kassel.de