Cleaning out the junk

[Jonathan Post]

It might be nice to have at least p_3 in OEIS, with a comment showing
some of the infinite array A[k,n] = p_k(n). Plus the crossreferences
which you've identified.



Leroy,

Using this PARI code:

\p 4444;
a = vector(20); a[1]=2;
fx(n) = prod(j=1,n-1, 1/(1 +1/a[j]));
fy(n) = sum(k=1,n-1, 1/a[k]);
fa(n) = ceil((1 + y(n) + sqrt((y-1)^2 + 4*x))/(2*(x-y)));
for(n=2,20,x = fx(n); y = fy(n); a[n] = fa(n));

I was able to calculate terms up to n=13 (probably could do more but 
they're huge).
The difference between the sum and product for the terms up to n=13 is 
9.7162... E-2885.
The constant (assuming convergence) begins as
.603778945883813107507744027627091423534456195946805046549148681848593524...
with a continued fraction
[0, 1, 1, 1, 1, 9, 1, 65, 2, 1864, 1, 5, 1, 1, 1, 60703, 1, 1, 6, 1, 1, 6, 
5, 2, 1, 35, 2, 20731941, 16, ...
The constant isn't in Plouffe's Inverter.

The beginning terms are
2, 10, 265, 186534, 39698716206, 9708281043219621795399, 
485147416562376967927656482516055847985046599, 
261312356099926248292437979417147998592741394591619008401746229884484893481820640113595606

Gerald McGarvey

At 11:46 AM 11/6/2007, Leroy Quet wrote:
>Regarding sequence A134473:
>
> >%I A134473
> >%S A134473 2,10,265,186534
> >%N A134473 a(n) = the smallest positive integer
>such that sum{k=1 to n}
> >1/a(k) is <= product{j=1 to n} 1/(1 +1/a(j)),
>for every positive integer n.
> >%C A134473 sum{k=1 to n} 1/a(k)  increases, but
>is bounded from above (by
> >the product). While product{j=1 to n} 1/(1
>+1/a(j)) decreases, and is
> >bounded from below (by the sum). The sum and the
>product then approach the
> >same constant, which is approximately
>..6037789..., if their difference
> >approaches 0. Does this constant have a closed
>form in terms of known
> >constants, if the constant exists?
> >%F A134473 For n >= 2,
> >if x = product{j=1 to n-1} 1/(1 +1/a(j)), and y
>= sum{k=1 to n-1} 1/a(k),
> >then
> >a(n) = ceiling[(1 + y + sqrt((y-1)^2 +
>4x))/(2(x-y))].
> >%e A134473 sum{k=1 to 2} 1/a(k) = 3/5, and
>product{j=1 to 2} 1/(1 +1/a(j))
> >= 20/33. For m = any positive integer <= 264,
>3/5 + 1/m is >
> >20/33/(1 + 1/m). But if m = 265, then 3/5 + 1/m
>= 32/53 is <= 20/33/(1
> >+ 1/m) = 2650/4389. So a(3) = 265.
> >%Y A134473 A134474,A134475,A134476,A134477
> >%O A134473 1
> >%K A134473 ,more,nonn,
>
>
>I haven't thought too hard about this, but it
>seems that it would be easy to prove that the sum
>and the product approach the same constant. (The
>two limits are mighty close, whatever the
>situation.)
>
>Call the sum limit S and the product limit P.
>
>In any case, do the constants S and P
>(.6037789... = the continued fraction
>[0;1,1,1,1,9,1,65,....]) have (a) closed form(s)
>in terms of known constants?
>
>
>Thanks,
>Leroy Quet
>
>
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