The (N+k)/k prime puzzle. If I've read it before I don't remember it!

Perhaps it was posted on this site a year ago or so when I was

visiting, and it lodged in my subconscious :)

I couldn't believe I sent the 12,252,240 + n is divisible by n thing.

I realized a minute after I sent it what I had done, but it was too

late!

The Nth number has N prime factors puzzle is brilliant.

That the Nth number is also divisible by N is not too improbable, on

inspection. The 2,3,6 8th and 9th numbers *must* be divisible by N.

Getting both the 5th and 10th numbers divisible by 5 and 10 is about

a four in five chance, assuming there is not some unknown principle

which might make it more certain or even guaranteed. Getting the

seventh number divisible by 7 is about fifty fifty, again assuming

there is not some unknown princple stacking the deck further.

Mark

--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"

<jens.k.a@g...> wrote:> Mark Underwood wrote:

program can easily

>

> > Fun with successive numbers:

> >

> >

> > 18,164,161 is a prime times 1

> ...

> > 18,164,167 is a prime times 7.

>

> 2,918,756,139,031,688,155,200 + k is a prime times k, for k = 1..14

> See http://www.primepuzzles.net/puzzles/puzz_181.htm

> 5,516,280 + k is the smallest solution for k = 1..7

>

> > 34,415,168 is divisible by 2

> > 34,415,169 is divisible by 3

> > 34,415,170 is divisible by 5

> > 34,415,171 is divisible by 7

> > 34,415,172 is divisible by 11

> > 34,415,173 is divisible by 13

> > 34,415,174 is divisible by 17

> > 34,415,175 is divisible by 19

> > 34,415,176 is divisible by 23.

>

> This is what CRT (Chinese Remainder Theorem) is for. A bigint

> reach primes in the millions but I will spare you solutions with

millions of

> digits.

12,252,240

>

> > 12,252,242 is divisible by 2

> ...

> > 12,252,258 is divisible by 18.

>

> Because lcm(2,...,18) = 2^4 * 3^2 * 5 * 7 * 11 * 13 * 17 =

> (lcm = least common multiple)

of 10

>

>

> My contribution to fun with successive numbers is the smallest case

> numbers where the nth has n prime factors:

case

>

> 3931520917431241 = 3931520917431241

> 3931520917431242 = 2 * 1965760458715621

> 3931520917431243 = 3 * 221477 * 5917124453

> 3931520917431244 = 2 * 2 * 23 * 42733923015557

> 3931520917431245 = 5 * 17 * 199 * 5557 * 41826179

> 3931520917431246 = 2 * 3 * 29 * 83 * 261799 * 1039837

> 3931520917431247 = 7 * 7 * 7 * 19 * 43 * 557 * 25187741

> 3931520917431248 = 2 * 2 * 2 * 2 * 31 * 59 * 167 * 804471071

> 3931520917431249 = 3 * 3 * 3 * 3 * 3 * 41 * 53 * 1721 * 4326271

> 3931520917431250 = 2 * 5 * 5 * 5 * 5 * 5 * 11 * 13 * 13 * 338377271

>

> The nth number is divisible by n above, but not in the 2nd smallest

> starting at 5,818,684,827,160,441.

>

> See http://www.research.att.com/projects/OEIS?Anum=A072875

>

> --

> Jens Kruse Andersen