[seqfan] Re: Question about limit of 1/(k*Pi(k))

[Frank Adams-Watters]

See A03810 and A038111 to see what is known.

Franklin T. Adams-Watters

-----Original Message-----
From: Bob Selcoe <rselcoe at entouchonline.net>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Thu, Jan 8, 2015 10:35 pm
Subject: [seqfan] Re: Question about limit of 1/(k*Pi(k))


Hi Seqfans,

Thanks again, Jack (and Olivier).

Ultimately, I'm trying to determine ratios of the natural numbers, by 
sets
whose smallest factor is Pi(k).

It is certainly the case that the sum of these ratios, whatever they 
are,
equals 1.

So for Pi(1) = 2, the set is the even numbers.  1/2 of the natural 
numbers
are even.

Pi(2) = 3, the set is odd multiples of 3 == 1/6  of the natural numbers.
Pi(3) = 5, the set is A084967:  {5, 25, 35, 55, 65, 85, 95, 115, 125, 
145,
155, 175...} == 1/15
Pi(4) = 7, the set is  A084968:  {7, 49, 77, 91, 119...}.  This appears 
to
be == 4/105.
Pi(5) = 11, the set is A084969.  The ratio is ???

Is there some equation for this?  A way to estimate?

Cheers,
Bob

--------------------------------------------------
From: "Jack Brennen" <jfb at brennen.net>
Sent: Thursday, January 08, 2015 11:29 AM
To: <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Question about limit of 1/(k*Pi(k))

> Here's your sum:
>
>    http://oeis.org/A124012
>
> Or 0.848969034043...
>
> - Jack
>
>
> On 1/8/2015 9:11 AM, Olivier Gerard wrote:
>> On Thu, Jan 8, 2015 at 5:10 PM, Bob Selcoe 
<rselcoe at entouchonline.net>
>> wrote:
>>
>>> Hello Seqfans,
>>>
>>> Is the following true, and if so, a known property?
>>>
>>> Where Pi(k) is the k-th prime and k = 1..inf:
>>>
>>> Sum_(1/(k*Pi(k))) = 1
>>>
>>> i.e., 1/2 + 1/6 + 1/15 + 1/28 + 1/55...
>>>
>>
>> No, this is not true, by a significant margin.
>>
>> This sum is strictly inferior to
>>
>> Sum_(1/(k*(k+1)))  which is equal to 1.
>>
>>
>> Olivier
>>
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>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
>
>
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