## Consecutive integers: 1*prime, 2*prime, 3*prime, 4*prime,5*prime,...

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• Andrey s line of questioning brought this (much simpler!) question to mind. We want to find consecutive integers: 1*prime,2*prime, 3*prime,
Message 1 of 6 , Feb 25, 2012
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Andrey's line of questioning brought this (much simpler!) question to mind.

We want to find consecutive integers: 1*prime,2*prime, 3*prime, 4*prime,5*prime,...

How far can you go? I got to three using an advanced carbon based, parallel bioprocessor running at about 15 Hz:

1*13, 2*7, 3*5.

Mark
• ... Somebody once posted this problem here. Guess who! http://tech.groups.yahoo.com/group/primenumbers/message/16364 It s also at
Message 2 of 6 , Feb 25, 2012
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Mark wrote:
> We want to find consecutive integers: 1*prime,2*prime, 3*prime,
> 4*prime,5*prime,...

Somebody once posted this problem here. Guess who!

It's also at http://www.primepuzzles.net/puzzles/puzz_181.htm
and http://oeis.org/A074200

--
Jens Kruse Andersen
• Good lord! It s rather disconcerting that I have absolutely no memory of that. Jens, I admire your power of recall. I think I ve regressed, hehe! Mark
Message 3 of 6 , Feb 25, 2012
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Good lord! It's rather disconcerting that I have absolutely no memory of that. Jens, I admire your power of recall. I think I've regressed, hehe!

Mark

--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen" <jens.k.a@...> wrote:
>
> Mark wrote:
> > We want to find consecutive integers: 1*prime,2*prime, 3*prime,
> > 4*prime,5*prime,...
>
> Somebody once posted this problem here. Guess who!
>
> It's also at http://www.primepuzzles.net/puzzles/puzz_181.htm
> and http://oeis.org/A074200
>
> --
> Jens Kruse Andersen
>
• ... --The number just before such an n-term sequence must be divisible by n!, according to my advanced bioprocessor. Say it is n!*k. Then n!*k+1=prime,
Message 4 of 6 , Feb 28, 2012
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--- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
>
> Andrey's line of questioning brought this (much simpler!) question to mind.
>
> We want to find consecutive integers: 1*prime,2*prime, 3*prime, 4*prime,5*prime,...
>
> How far can you go? I got to three using an advanced carbon based, parallel bioprocessor running at about 15 Hz:
>
> 1*13, 2*7, 3*5.
>
> Mark

--The number just before such an n-term sequence must be divisible by n!,
according to my advanced bioprocessor. Say it is n!*k.
Then n!*k+1=prime, n!*k/2+1=prime, n!*k/3+1=prime, n!*k/4+1=prime, etc.

I think this is a pretty good problem, actually. I can't see why it could not happen for arbitrarily large n, and furthermore, whenever it does happen, it'll be pretty easy to prove primality for the primes p, since p-1 is easily factored.
• ... Not exactly so; it only needs to be divisible by least common multiple of numbers [1..n]. For example, for n = 6, the first occurrence is V = 5516280 =
Message 5 of 6 , Feb 28, 2012
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> --The number just before such an n-term sequence must be divisible by n!,
> according to my advanced bioprocessor. Say it is n!*k.
> Then n!*k+1=prime, n!*k/2+1=prime, n!*k/3+1=prime, n!*k/4+1=prime, etc.

Not exactly so; it only needs to be divisible by least common multiple of
numbers [1..n]. For example, for n = 6, the first occurrence is

V = 5516280 = 60*91938 = 720*7661.5
V+1 = 1*5516281
V+2 = 2*2758141
V+3 = 3*1838761
V+4 = 4*1379071
V+5 = 5*1103257
V+6 = 6*919381

Peter
• right, good correction.
Message 6 of 6 , Feb 28, 2012
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right, good correction.

>Not exactly so; it only needs to be divisible by least common multiple of numbers [1..n].
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