Lengths Of Runs In The #-Of-Divisors Sequence

[Warut Roonguthai]

On 7/17/07, Leroy Quet <qq-quet at mindspring.com> wrote:
>
> It seems VERY likely to me that there is no infinite string of 1's, or of
> anything else, in sequence A131789 (ie. the terms of A131790 are all
> finite).
>
> Can it be PROVED that all terms of A131790 are finite, possibly using
> Hardy and Wright or some other such reference?

Since there are infinitely many primes, 1 will appear infinitely often
in A131789. But there cannot be an infinite string of 1's in A131789
because there exist infinitely many m such that d(m) = d(m+1) as
proved by Roger Heath-Brown in 1984. So all terms of A131790 are
finite.

Warut



Simon Plouffe:

> And for the number 339066 the prime generated has 255 digits

Looking at just the record values, from here on...

{record-index, seed-number, digits-in-target-prime}

{41,339066,255}
{42,594594,317}
{43,902538,328}
{44,1750014,341}
{45,2254098,346}
{46,3174138,352}
{47,3467646,354}
{48,3818178,446}
{49,3913434,447}
{50,8795358,501}
{51,9489018,502}
{52,9522414,503}
{53,13891878,511}
{54,14139762,514}
{55,14167494,515}
{56,19803966,522}
{57,23978262,529}
{58,24478146,594}
{59,28289898,673}
{60,35837802,1025}




I'm adding Eric's two new sequences: they will
be A120712 and A120713.  I used Simon Plouffe's 
nice Maple program to generate them, but modified
it to give exactly the same terms as Eric had.

To get an analogue of the home primes, A037274, I suppose
we could iterate the map (k -> concatenation of proper
divisors of k) until we reach a prime; then a(n) would
be the prime we eventually reach when starting with n,
or -1 if we never reach a prime.  But what about a(p)
where p is prime?  I guess we set that to equal p.

And we can set a(1) = 1.  So the sequence starts
1 2 3 2 5 23 7 ...
What is a(8)?
I don't know, but I will put it in the OEIS as A120716.
Hopefully someone will extend it.

Neil




Seqfans,  for the new sequence A120716 that I mentioned,
the analogue of home primes, the big question is,
what is a(8)?
we have 

and the divisors of that last number aree

1, 2, 3, 6, 37, 74, 111, 113, 222, 226, 339, 678, 4181, 8362, 12543, 25086,
3192525397, 6385050794, 9577576191, 19155152382, 118123439689, 236246879378,
354370319067, 360755369861, 708740638134, 721510739722, 1082266109583,
2164532219166, 13347948684857, 26695897369714, 40043846054571, 80087692109142,
30132785470246166539693, 60265570940492333079386, 90398356410738499619079,
180796712821476999238158, 1114913062399108161968641, 2229826124798216323937282,
3344739187197324485905923, 3405004758137816818985309, 6689478374394648971811846,
6810009516275633637970618, 10215014274413450456955927,
20430028548826900913911854, 125985176051099222302456433,
251970352102198444604912866, 377955528153297666907369299,
755911056306595333814738598, 96199682896113474519861511083121,
192399365792226949039723022166242, 288599048688340423559584533249363,
577198097376680847119169066498726, 3559388267156198557234875910075477,
7118776534312397114469751820150954, 10678164801468595671704627730226431,
10870564167260822620744350752392673, 21356329602937191343409255460452862,
21741128334521645241488701504785346, 32611692501782467862233052257178019,
65223385003564935724466104514356038, 402210874188650436967540977838528901,
804421748377300873935081955677057802, 1206632622565951310902622933515586703,
2413265245131902621805245867031173406 ]

If we throw away the first and last terms and concatenate the rest,
is that a prime?  Does someone have access to a really good
prime tester?!

Neil

PS I did not check these calculations and it is 05:00