[seqfan] Sequence with unique geometric means

Jack Brennen jfb at brennen.net
Fri Apr 30 19:51:07 CEST 2010


I thought of an interesting sequence and computed a few terms...

The sequence is chosen greedily, with each subsequent term being
the smallest that can be added such that no two non-empty subsets
of the sequence have equal geometric mean.

The first term being 1, the sequence looks like this, going out
for the first 51 terms:

   1, 2, 3, 5, 7, 8, 11, 13, 17, 18, 19, 23, 29, 31, 37, 41,
   43, 47, 50, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98,
   101, 103, 107, 109, 113, 127, 131, 137, 139, 149,
   151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Obviously it contains all of the primes, but it also contains
a sprinkling of composites.  Among the first 51 terms, we only
have 4 composites:  8, 18, 50, and 98.  Perhaps not coincidentally,
these are the first four composites of the form 2p^2.

Also, it's interesting that this sequence is very similar in the
beginning to the sequence A066720.  It doesn't deviate until the
22nd term.  A066720 includes the number 60, while this sequence
does not because these two sets have the same geometric mean:

   {2,3,5,8,18,50}
   {1,60}

Later, A066720 includes the number 81, which isn't here because
these have the same geometric mean:

   {2,5,8,18,50,81}
   {1,8,50,81}

It's intriguing to think that this new sequence might be a
subsequence of A066720, but since each sequence depends on
all previous elements of the sequence, it's likely that as
the sequences diverge, they start to look less and less alike...





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