[seqfan] Where are the primes?

[Eric W. Weisstein]

On Tue, 12 May 1998, Christian G.Bower wrote:

> Sloane's sequences A007908 (1, 12, 123, 1234, ...,) and A000422 (1, 21,
> > 321, 4321, ...) contain surprisingly few primes.
> 
> If you look at the sums of the digits of the sequence you find
> 1+2 is a multiple of 3
> 1+2+3 is a multiple of 3
> 1+2+3+4+5 is a multiple of 3
> 1+2+3+4+5+6 is a multiple of 3
> 1+2+3+4+5+6+7+8 is a multiple of 3
> 
> If the sum of the digits is divisible by 3, so is the number, so every
> third number is not divible by 3 and has a chance at being prime.  Every
> other one of these is even. So you can only find primes at every 6th
> number.  Since each trial adds at least 6 digits (soon 12, then 18...) 
> the numbers get big quickly so primes are sparse. 
> 
> This should at least explain why primes should be hard to find

Thanks; very true.  However, at least for my satisfaction, more
explanation is still needed.  The numbers don't seem grow so fast that you
would expect to find _so few_ primes (and I've now confirmed there are
none in the first 2350 terms...)  Similarly, the only prime in the first
2000 terms of the sequence 1, 21, 321, 4321, ..., is 828180...321

- -Eric

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