## puzzle: what is special about these 10 consecutive primes?

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• Puzzle: What is special about these 10 consecutive primes? 5981567 5981579 5981609 5981629 5981641 5981659 5981663 5981669 5981681 5981683 Mark
Message 1 of 6 , Jul 22, 2012
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Puzzle: What is special about these 10 consecutive primes?

5981567
5981579
5981609
5981629
5981641
5981659
5981663
5981669
5981681
5981683

Mark
• I don t know. So what is so special about your 10 sequential prime numbers? Sent from my iPad ... [Non-text portions of this message have been removed]
Message 2 of 6 , Jul 22, 2012
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I don't know. So what is so special about your 10 sequential prime numbers?

On 22 Jul 2012, at 15:54, "Mark" <mark.underwood@...> wrote:

>
> Puzzle: What is special about these 10 consecutive primes?
>
> 5981567
> 5981579
> 5981609
> 5981629
> 5981641
> 5981659
> 5981663
> 5981669
> 5981681
> 5981683
>
> Mark
>
>
>
>
>
>
>

[Non-text portions of this message have been removed]
• ... Solved within a minute by searching the first term in OEIS: http://oeis.org/A214219 That may seem like cheating but this property looks difficult to guess.
Message 3 of 6 , Jul 22, 2012
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Mark wrote:
> Puzzle: What is special about these 10 consecutive primes?
>
> 5981567
> 5981579
> 5981609
> ...
> 5981683

Solved within a minute by searching the first term in OEIS:
http://oeis.org/A214219
That may seem like cheating but this property looks difficult to guess.

So for each k from 1 to 10, k divides the sum of the
k consecutive primes starting at 5981567:
1 divides 5981567
2 divides 5981567 + 5981579
3 divides 5981567 + 5981579 + 5981609
...
10 divides 5981567 + ... + 5981683.

A214219 and http://www.primepuzzles.net/puzzles/puzz_415.htm
also give two more terms:
148871869 for k = 1 to 11, and 5545986967 for k = 1 to 12.
My computation agrees and shows 28511128379 for k = 1 to 13,
and 85185688439 for k = 1 to 14.

--
Jens Kruse Andersen
• Bingo! Yes with fresh eyes I see that the puzzle was rather obscure. Should have given a hint. But golly, is there anything that OEIS doesn t have? Good
Message 4 of 6 , Jul 22, 2012
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Bingo! Yes with fresh eyes I see that the puzzle was rather obscure. Should have given a hint. But golly, is there anything that OEIS doesn't have? Good find. Jens how you got up to 14 so quickly is beyond me (as usual!).

The investigation started out innocently enough, as I wanted to know when 2+3+5+7+... would first yield an average number that was an integer (other than 2!). As it turns out one must get to the 23rd prime, 83, for the average to come out to an integer (38). Then one must sum all the primes up to 241 to get the next integer average (110), and then up to the prime 6599 to get the next integer average (3066).

This is a very low yield compared to starting at other primes, at least most other primes anyways. I would expect the yield to be about half as much as average, but it seems less.

Mark

--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
>
> Mark wrote:
> > Puzzle: What is special about these 10 consecutive primes?
> >
> > 5981567
> > 5981579
> > 5981609
> > ...
> > 5981683
>
> Solved within a minute by searching the first term in OEIS:
> http://oeis.org/A214219
> That may seem like cheating but this property looks difficult to guess.
>
> So for each k from 1 to 10, k divides the sum of the
> k consecutive primes starting at 5981567:
> 1 divides 5981567
> 2 divides 5981567 + 5981579
> 3 divides 5981567 + 5981579 + 5981609
> ...
> 10 divides 5981567 + ... + 5981683.
>
> A214219 and http://www.primepuzzles.net/puzzles/puzz_415.htm
> also give two more terms:
> 148871869 for k = 1 to 11, and 5545986967 for k = 1 to 12.
> My computation agrees and shows 28511128379 for k = 1 to 13,
> and 85185688439 for k = 1 to 14.
>
> --
> Jens Kruse Andersen
>
• Nice . Those interested in this puzzle , or even if not , may be interested in Doric Columns of Primes , http://upforthecount.com/math/pdor.html There are some
Message 5 of 6 , Jul 22, 2012
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Nice .

Those interested in this puzzle , or even if not , may be
interested in Doric Columns of Primes ,
http://upforthecount.com/math/pdor.html
There are some "columns" there , some questions and a
brief description of programming which produced massive
overlap between integer and floating-point processor
utilization .

Cheers ,

Walter
• ... Your text was written in 1991. http://www.primepuzzles.net/puzzles/puzz_181.htm from 2003 has the same 11 terms up to 3163427380990800 and adds 3 more
Message 6 of 6 , Jul 23, 2012
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Walter Nissen wrote:
> Those interested in this puzzle , or even if not , may be
> interested in Doric Columns of Primes ,
> http://upforthecount.com/math/pdor.html

Your text was written in 1991.
http://www.primepuzzles.net/puzzles/puzz_181.htm from 2003 has the
same 11 terms up to 3163427380990800 and adds 3 more terms:
12 22755817971366480 Phil Carmody
13 3788978012188649280 " "
14 2918756139031688155200 J. K. Andersen (20/4/03)