[seqfan] Re: A045575

[M. F. Hasler]

On Mon, Jun 12, 2023 at 10:41 AM Hans Havermann <gladhobo at bell.net> wrote:

> https://oeis.org/A045575

THANKS !

> I recently calculated that there are 133 090 654 terms of this sequence
> less than 10^60000. The above Pillai formula suggests 6.81215*10^7, unless
> I miscalculated.


FWIW I confirm this value, *assuming that log is the log to base e*.
To double-check, to avoid overflow I define the function in terms of the
exponent X:
L(X) = log(X*log(10)) /* this is log(10^X) */
N(X) = .5*L(X)^2/log(L(X))^2 /* this is actually N(10^X)*/
N(6e4)
68 121 509.310579961606253428301636462648

So this is half of your value...
When I compute  N( a(10^4) )  with Pillai's formula and  a(10^4) ~
10^218.224  from the b-file,
I get  3263.5,  which is off by a factor 3 from the exact value of 10^4,
assuming the b-file is correct.

So, either the convergence is very slow (off by factor 3 at 10^218, off by
factor 2 at 10^60000)
and/or there is an error with the "log"...?
If I take  log = log_10 instead, then I get :
NN(218)  = 4345.3...
NN(60000)  = 78841107.995...
Slightly closer - but we see that for 60 000 it doesn't change much,
and it's actually easy to see that the base of the log doesn't matter,
asymptotically:
it's just a multiplicative factor that completely simplifies for the outer
logs,
and that becomes just an added constant after taking the second log in the
denominator.

So, finally I guess that it's just a matter of very slow convergence to the
asymptotic expression.

- Maximilian