[seqfan] Re: 'Climb to a prime' in other bases

[Neil Sloane]

David Seal:  Nice work!

It turns out that the base-2 version is already in the OEIS,
although without your counterexample, which I have now added.

I also added some sequences that were missing, so the base-2 problem is now
described in the following 5 entries:
A230625 & A287874 for the basic map,
and, for what you reach when the map is
iterated, A230626, A230627, A287875.

There is a slight awkwardness in A287875, because the escape clause, which
says we write -1 if we don't reach 1 or a prime, is tricky to deal with in
base 2.  So the escape clause is "a(n) = -1 in decimal if ...", and
otherwise "a(n) = the prime reached, or 1, written in binary".

Did you check to see if 255987 is the smallest number that is fixed in base
2?  I assume so, but it wasn't clear from your message.

It would be interesting to see the list of numbers that don't reach 1 or a
prime.  That is, the numbers that are fixed (and composite), or go to a
composite fixed point, or go into a loop.  If you have this list, could you
submit it as a new sequence?


Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Tue, Jun 13, 2017 at 9:31 AM, David Seal <david.j.seal at gwynmop.com>
wrote:

> The recent posts about 13532385396179, http://oeis.org/A080670 and
> http://oeis.org/A195264 caused me to wonder whether similar numbers exist
> when you write them in bases other than 10. I've done some computer
> investigations of the question, so far only using 'brute force' programs
> (I've got a number of ideas how to improve on them), with the following
> results:
>
> First, there are a couple of very simple examples in bases 6 and 8:
>
> base-6 24 = decimal 16 = decimal 2^4 = base-6 2^4
>
> base-8 33 = decimal 27 = decimal 3^3 = base-8 3^3
>
> More generally, for any prime p and integer n > 0, the 2-digit number with
> digits p, np is equal to p^np in base b = p^(np-1)-n. That doesn't lead to
> a solution for the (p,n) = (2,1) combination, since that implies b=1 and so
> has the problem that the digits are >= the base, but otherwise it works,
> leading to a doubly-infinite family of bases in which examples exist, the
> next one of which is 29:
>
> base-29 26 = decimal 64 = decimal 2^6 = base-29 2^6
>
> Other than that, I have tried all bases from 2 to 20, and found examples
> in bases 2, 11 and 12:
>
> binary 111110011111110011
> = decimal 255987
> = decimal 3^3 * 19 * 499
> = binary 11^11 * 10011 * 111110011
>
> base-11 3518
> = decimal 4617
> = decimal 3^5 * 19
> = base-11 3^5 * 18
>
> base-12 15287
> = decimal 29767
> = decimal 17^2 * 103
> = base-12 15^2 * 87
>
> In binary, I've also found a couple of 'amicable pairs', in which the
> operation of writing down the factorisation and then concatenating the
> primes and exponents on each number in the pair produces the other one. The
> first is:
>
> binary 1111101111
> = decimal 1007
> = decimal 19 * 53
> = binary 10011 * 110101
>
> binary 10011110101
> = decimal 1269
> = decimal 3^3 * 47
> = binary 11^11 * 101111
>
> and the second is:
>
> binary 10111011111
> = decimal 1503
> = decimal 3^2 * 167
> = binary 11^10 * 10100111
>
> binary 111010100111
> = decimal 3751
> = decimal 11^2 * 31
> = binary 1011^10 * 11111
>
> And the first of those two happens to work in base 4 as well:
>
> base-4 33233
> = decimal 1007
> = decimal 19 * 53
> = base-4 103 * 311
>
> base-4 103311
> = decimal 1269
> = decimal 3^3 * 47
> = base-4 3^3 * 233
>
> So the 'climb to a prime' conjecture is false in bases 2, 4, 6, 8, 10, 11,
> 12, 29 and the various higher bases of the form p^(np-1)-n with p prime, n
> >= 2.
>
> There are also various negative results - these are obviously dependant on
> there being no as-yet-undetected bugs in the programs I've written. They
> search for length-L "loops" in iterative application of the operation,
> where a number produces itself after L applications of the operation (so
> the 'amicable pairs' are length-2 loops and the other examples above are
> length-1 loops). More specifically, they search for non-prime maximal
> elements of loops, which are obviously in 1-to-1 correspondence with loops
> other than the length-1 loops that contain a prime number only. The results
> are:
>
> Bases 2-6: No non-prime maximal elements of loops found other than those
> listed above (1269, 3751 and 255987 in base 2; 1269 in base 4; 16 in base
> 6) up to 6.5*10^8. (Programs are still running, so that limit will probably
> increase over time.)
>
> Bases 7-20: No non-prime maximal elements of loops found other than those
> listed above (27 in base 8; 4617 in base 11; 29767 in base 12) up to 7^10,
> 8^10, 9^9, 10^9, 11^8, 12^8, 13^8, 14^7, 15^7, 16^7, 17^7, 18^7, 19^7, 20^6
> respectively. (These results were obtained by an earlier, memory-bound
> version of the program, which I am no longer running.)
>
> There's obviously the beginnings of an integer sequence here, defined as
> the smallest maximal element of a sequence generated by the 'climb to a
> prime' operation in base N that doesn't actually climb to a prime, or -1 if
> no such sequence has a maximal element. But 1269, ?, 1269, ?, 16, ?, 27, ?,
> ?(<=13532385396179), 4617, 29767, ... is probably a bit too little
> knowledge to register an OEIS sequence on!
>
> David Seal
>
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