[seqfan] Re: A209270 vs A083698

Hans Havermann gladhobo at bell.net
Fri Dec 27 00:16:03 CET 2019


Convergents[{2,1,1,2,2,4,6,8,4,6,38}]
{2,3,5/2,13/5,31/12,137/53,853/330,6961/2693,28697/11102,179143/69305,6836131/2644692}

Convergents[{0,2,1,1,2,2,4,6,8,4,6,38}]
{0,1/2,1/3,2/5,5/13,12/31,53/137,330/853,2693/6961,11102/28697,69305/179143,2644692/6836131}

Let me see if I understand this. To make sense of the A083698 definition, I have to realize that there is a 0 hidden before the initial term of the sequence? In some unknown-to-me formalism, I suppose that might at least explain why the offset was chosen to be one, not zero.


> On Dec 17, 2019, at 3:54 PM, Tomasz Ordowski <tomaszordowski at gmail.com> wrote:
> 
> Hans,
> 
> Note that 1 / [a(1);a(2),...,a(n)] = [0;a(1),a(2),...,a(n)].
> 
> Thomas
> 
> wt., 17 gru 2019 o 20:11 Hans Havermann <gladhobo at bell.net> napisał(a):
> 
>> I haven't seen any movement on this issue other than M.F. Hasler editing
>> A209270 with a "same as A083698" crossref.
>> 
>> I will point out that the Mathematica "Convergents" function of the
>> initial terms yields {2, 3, 5/2, 13/5, 31/12, 137/53, 853/330, 6961/2693,
>> 28697/11102, 179143/69305, ...}, so I can see that the numerators are in
>> fact the prime terms of A072999 which (although A072999 isn't mentioned in
>> A209270) reflects the wording of Benoit Cloitre's duplicate version.
>> 
>> Paul Hanna's earlier A083698 has: "Partial quotients of the continued
>> fraction which has convergents with the least possible prime denominators
>> (A072999)." In light of the aforementioned convergents list, I am not
>> really understanding this. Is this simply a case of the word "denominators"
>> being incorrect?





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