[QUOTE=3.14159;228784]Strangely, Benford's law did not kick in. (The leading digit should normally be 1.)[/QUOTE]

You expect an initial 9 about 4.6% of the time, so it's not that unusual. To make it seem stranger you could use a base 100 Benford's law... :smile:

[QUOTE=3.14159;228784]Did you submit anything for the only entry you seemed to like, item 16? Number, square, and fourth?[/QUOTE]

No, but not for lack of trying. I spent about 40 processor-hours looking for big examples in promising places.

[QUOTE=Charles]You expect an initial 9 about 4.6% of the time, so it's not that unusual. To make it seem stranger you could use a base 100 Benford's law...
[/QUOTE]

Yes, but 1 is expected about 33% of the time under Benford's law.

[QUOTE=Charles]No, but not for lack of trying. I spent about 40 processor-hours looking for big examples in promising places.
[/QUOTE]

.. Numbers with many divisors?

And, speaking of Benford's law: The largest prime I have ever found as of yet follows Benford's law. It begins with 1057524716542310847527293760394460309343436182462...

PRP: 4581 * 2[sup]45720[/sup] + 1;

Submission for category 20!

[code]4581*2^45720 + 1 is prime! (a = 5) [13767 digits]
4581*2^45720 + 1 is prime! (verification : a = 11) [13767 digits][/code]

Sadly, it did not divide any Fermat or Generalized Fermat number, or any other property of the sort. Normal proth number.

[code]4581*2^45720 + 1 doesn't divide any Fm.
4581*2^45720 + 1 doesn't divide any GF(3, m).
4581*2^45720 + 1 doesn't divide any GF(5, m).
4581*2^45720 + 1 doesn't divide any GF(6, m).
4581*2^45720 + 1 doesn't divide any GF(10, m).
4581*2^45720 + 1 doesn't divide any GF(12, m).
4581*2^45720 - 1 factor : 5
4581*2^45721 + 3 factor : 3
4581*2^45721 + 1 factor : 7
4581*2^45719 + 1 is composite. (a = 7)[/code]

[QUOTE=3.14159;228789]Yes, but 1 is expected about 33% of the time under Benford's law.[/QUOTE]

I get 30% for base-10 numbers. 33% is more like base-8.

You could look through your non-base-related record primes and see how many start with each digit, then do a chi-square to see if anything funny is going on. You wouldn't expect too many or too few of any initial digit. (If 3 out of your 9 primes start with 1, that's usual; if 3,000,000 out of your 9,000,000 primes start with 1, that's too many.)

To be random: Is there any specific function to determine the ratio of integers not divisible by the first n primes?

[QUOTE=3.14159;228803]To be random: Is there any specific function to determine the ratio of integers not divisible by the first n primes?[/QUOTE]

I have a thread on Math addressing this. It's closely approximated by
[TEX]\frac{e^{-\gamma}}{\log n+\log\log n}[/TEX]
assuming you mean "not divisible by any of the first n primes".

By gamma, do you mean, 0.5772156649015328.. ?

[QUOTE=3.14159;228807]By gamma, do you mean, 0.5772156649015328.. ?[/QUOTE]

Yes. This is (one form of one of) Mertens' Law(s).

[QUOTE=3.14159;228577][B]General numbers only. [/B][/QUOTE]

So what does this mean? The last time I asked about this (on the other thread, I think) you didn't give me a satisfactory definition. Apparently Fermat numbers don't count, but what else?

[QUOTE=Charles]So what does this mean? The last time I asked about this (on the other thread, I think) you didn't give me a satisfactory definition. Apparently Fermat numbers don't count, but what else?
[/QUOTE]

Any random prime integer. If you'd like, it can also be anything from the list.

I disallow Fermats and Mersennes because they have special-form factors (The former is 2kp + 1, the latter are Proth numbers.)

Example: 636678617124118560273979 = 9978 * 1999[sup]6[/sup] + 1. This number is allowed.

Example of a disallowed number: 4435164151219413385217 = 508[sup]8[/sup] + 1. This is disallowed due to special-form potential divisors, which make it too easy.