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What is the limit of the continued fraction whose partial denominators are the composites?

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The posting says "below", but it isn't there. Let's see how it looks here: $1+\cfrac{1}{4+\cfrac{1}{6+\cfrac{1}{8+\cfrac{1}{9+ \cfrac{1}{10+\cfrac{1}{12+\cfrac{1}{14+\cfrac{1}{15+\cdots}}}}}}}}$ –  Michael Hardy Nov 18 '11 at 1:58
    
This question is the dual of this question. –  J. M. Nov 18 '11 at 2:33
    
Maybe I should have omitted "$1+{}$" at the beginning. –  Michael Hardy Nov 18 '11 at 2:55

2 Answers 2

You mean $1/(1+1/(4+1/(6+1/(8+1/(9+1/\ldots)$? It is approximately

1.240193472713540402907659901653619438804869904021954760571414246112567530857899396707443536897205404

but I doubt that there is a "closed form".

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Actually your numerical quote is for the reciprocal of the continued fraction. –  Sasha Nov 17 '11 at 21:01
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The Plouffe inverter pi.lacim.uqam.ca/eng recognizes neither this number nor its reciprocal. –  Gerry Myerson Nov 17 '11 at 22:55

That the continued fraction converges can be easily established through the Śleszyński-Pringsheim theorem.

For reference, here's some Mathematica code for computing this number to prec significant figures, which is similar to the code given here:

Composite[n_Integer?Positive] := FixedPoint[(n + PrimePi[#] + 1) &, n]

prec = 500;
y = N[1, prec + 5]; c = y; d = 0; k = 1;
While[True, p = Composite[k];
  c = p + 1/c; d = 1/(p + d);
  h = c*d; y *= h;
  If[Abs[h - 1] <= 10^(-prec), Break[]];
  k++];
N[y, prec]

which yields

1.2401934727135404029076599016536194388048699040219547605714142461125675308578993967074435368972054040499444569870631879216700131159033931437886878647547511888183754244113415120681779806667661819237665606323444679166964461726592199240306129802843968806256258252294199900746887849052219088971773650043418994122502389225153031976851620685269261818925150482368821451503313305347497194089489582641069356680422299551667227442255378383831719803128578492721019246089346156505612206821587675561358847518295030

Its reciprocal is

0.80632580480527957525355985703886211400308652751886019336591742291710688332574070079436098605441519195401666000252471896335967661438294029308273049422789739675889321789197956373719153024896914087422889469613361550097143208723090059691265863842011401757403471316704470493263754267277999620082114009184258161800044132073167763756696153308835343268670549110722478950038217797937443267139158452067897088395434297330559287749430628860526069885283389279384016537671968819880420551471111687607529969189158471


Gerry has said that Plouffe's inverter can't recognize these numbers; neither can the ISC.

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I didn't say Plouffe's inverter doesn't work; I just said it doesn't recognize either of these numbers. –  Gerry Myerson Nov 18 '11 at 3:38
    
I mean it doesn't work on these numbers. But sure, I'll clarify... –  J. M. Nov 18 '11 at 3:45

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