[seqfan] A130011 and the definition of "slowest increasing".

[Lars Blomberg]

Hello Seqfans,

 

A130011 has the terms:

a: 1, 4, 5, 12, 15, 16, 17, 18, 19, 20, 21, 36, 37, 38, 45, 48, 51, 54, 57,
60, 63, 64, 65, 66, 67, 68, 69, ...

Recently, I proposed (as did Nathaniel Johnston in 2011)

b: 1, 4, 5,  6, 13, 16, 19, 20, 21, 22, 23, 24, 25, 40, 41, 42, 49, 50, 51,
58, 61, 64, 67, 70, 73, 76, 77, ...

 

Both sequences fulfill the requirement that for any n in the sequence "There
are n terms in the sequence which are <= 3n".

But the second one is not accepted because "Alois P. Heinz: You have
b(27)=77, but this sequence has a(27)=69, which is smaller."

Yes, a(27) < b(27), but what about a(4) > b(4), a(12) > b(12), etc?

 

Nathaniel Johnston makes the change from b to a with the comment 

"... somewhat surprisingly (to me) the greedy algorithm does not produce the
optimal sequence."

But what, excactly, makes a the "optimal sequence".

 

Could someone please define what "slowest increasing" means?

And what is the difference between "slowest increasing" and
"lexicographically first"?

 

Best regards,

Lars B