# [seqfan] Re: Where are the primes?

Christian G.Bower bowerc at usa.net
Tue May 12 04:21:10 CEST 1998

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

> On another curious note, the Smarandache Function (The smallest value S(n)
> for a given n for which n|S(n)! ;
> not attain the following values for n=1 to 40,000: 24, 27, 30, 32, 36, 40,
> 42, 45, 48, 49, 50, 54, 55, 56, 60, ...

S(n) will eventually reach every number. The smallest k: S(k)=n is always
a prime power. The following method finds such a k given n. Let's say
n=24:

The prime factorization of 24 is 2^3 * 3

23! is divisible by 2^19 and 3^9, so for S(k) to be 24,
either 2^20|k or 3^10|k and the least of these to values is the
least values (3^10=59049, 2^20=1048576) is the least k such that
S(k)=24. This method works for all n:

24 : 59049       (3^10)
25 : 3125        (5^5)
26 : 169         (13^2)
27 : 177147      (3^11)
28 : 2401        (7^4)
29 : 29
30 : 78125       (5^7)
31 : 31
32 : 134217728   (2^27)
33 : 1331        (11^3)
34 : 289         (17^2)
35 : 16807       (7^5)
36 : 43046721    (2^33)

etc

Christian

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