# Tables of Permutation of Natural Numbers

Ralf Stephan ralf at ark.in-berlin.de
Fri Apr 18 12:27:34 CEST 2003

```Paul D. Hanna wrote
> Is the following conjecture a known theorem?
>[...]
> -----------------------------------------------------------------
>
> EXAMPLE 1: x=3/2.
>
> New EIS entry A083044:
> Square table read by anti-diagonals forms a permutation of the natural
> numbers:
> T(n,0)=floor(n*x/(x-1))+1, T(n,k+1)=ceil(x*T(n,k)),
> where x=3/2, n>=0, k>=0.
>[...]
> Table begins:
> 1  2  3  5   8   12  18  27  41  62   93   140 ...
> ...
> First row is A061419,

which, coincidentally, is one possible partial sum sequence of A073941,
which itself seems to have a connection to the Josephus problem, please see
also the sent-in updates to the entry (and I must apologize for my
recent comments having year 2002, won't happen again):

%Y A073941 Is this the same as A005428? - Benoit Cloitre, Nov 24, 2002
%C A073941 Same as log2(A082125(n)), for n>2?. - Ralf Stephan (ralf at ark.in-berlin.de), Apr 16 2003
%C A073941 Partial sums for various start indices seem to be in A006999,
A061419, A061418.

The formula
%F A073941 a(n) = ceiling(c*(3/2)^n-1/2) where c = 0.3605045561966149591015446628665... - Benoit Cloitre, Nov 22, 2002
I think this one is interesting,
- is there an easy ogf?
- confirming all digits of c, the ISC could not find it, does
it have a closed form?

Thanks,
ralf

```