Mian-Chowla sequence, A051788

[Pfoertner, Hugo]

Jens, Seqfans,

based on some experimentation my guess is, that there will be indeed
remaining gaps in the list of positive differences between the
elements of of A051788. A051788 has
494 entries below 2*10^6 and the list of differences not occurring
starts:
49 50 71 72 76 82 90 93 95 96 119 128 139 143 152 162 172 173
I checked that the entries below 128 are already present if one stops
searching
at 2*10^5 instead of 2*10^6.
So my guess is, that there is a new sequence: Differences not occurring
in A051788. I didn't check the situation for A005282, but it should be
similar,

Regards
Hugo
http://www.pfoertner.org/

-----Ursprüngliche Nachricht-----
Von: Pfoertner, Hugo 
Gesendet am: 04 February, 2003 10:38
An: 'jens at voss-ahrensburg.de'; Sequence Fanatics
Betreff: RE: Mian-Chowla sequence

The relevant sequence is A051788. The comment pointing to A005282 in
the A051788 sequence seems to be incorrect:

A variation on A005282 (Mian-Chowla, where positive _differences_ of
pairs of elements are unique) by starting with a(1) = 1, a(2) = 3
should read
A variation on A005282 (Mian-Chowla, where  _sums_ of
pairs of elements are unique) by starting with a(1) = 1, a(2) = 3

Hugo

-----Ursprüngliche Nachricht-----
Von: Jens Voss [mailto:jens at voss-ahrensburg.de]
Gesendet am: 04 February, 2003 08:26
An: Sequence Fanatics
Betreff: Mian-Chowla sequence


Hi fellow fanatics,

The Mian-Chowla sequence A005282 is defined as the sequence (a(n))
starting with 1 and where a(n) is the least value such that sequence
increases and such that pairwise sums of elements are all distinct.

An alternative definition is to start with 1 and then continue with the
least number such that all pairwise DIFFERENCES of distinct elements are
all distinct.

This second definition takes the focus to the differences of the members
of the Mian-Chowla sequence: Among the pairwise differences of the members
listed in the OEIS, the numbers 33, 39, 55, 61, 63, 87, 88, 91, 98, ...
do not occur, but it is very likely that at least some of them are
eliminated from this list further on (note that the difference of the last
two OEIS entries is only 72).

This observation of course provokes the following questions:
* Will all positive integers eventually end up as differences of members
  of A005282?
* If not, which ones (and how many) will remain?

Any input on this matter is welcome!

Regards,
Jens