[seqfan] Re: A181930

Benoît Jubin benoit.jubin at gmail.com
Tue Apr 3 02:38:57 CEST 2012

It appears that

sum( 1/p * product( (q-1)/q, q prime, q<p), p prime) = 1

since each summand is the asymptotic probability that the second
factor of a number be p. Since any number has a prime as second
factor, these probabilities add to 1.

Do you know a proof of this identity which does not use this
probability interpretation?

This sum converges very slowly (remainder ~ 1/ln(p) by the prime
number theorem).

PS.1: An alternative term is "asymptotic density of the set of natural
numbers with n-th factor equal to d" (see
PS.2: David, I added in A181930 the formula for T(d,tau(d)). Have you
submitted the associated sequence of "records" ?

On Mon, Apr 2, 2012 at 9:35 AM, David Wilson <davidwwilson at comcast.net> wrote:
> I have just edited and submitted A181930, which has to do with the
> probability that number d is the nth divisor of an integer.
> I added the "tabl" keyword, but I am not sure whether if it will format the
> table the way I would like.
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