# Near Integers

Artur grafix at csl.pl
Fri Jan 5 11:48:05 CET 2007

```Line: word divided was treated as divided modulo p (not exactly that
a!/p^m is
would be: word divided was treated as divided modulo p^m (not exactly that
a!/p^m is
ARTUR

Dnia 05-01-2007 o 11:27:56 Artur <grafix at csl.pl> napisał(a):

> Dear Max and Seqfans,
> Of course I'm agree that recent title is better. Also sign < have to be
> changed on <=.
> But I was interpreted previous title exactly follow Ribenboim book and
> word divided was treated as divided modulo p (not exactly that a!/p^m is
> integer)
> If we take a=6 and p=5 sample from the book formula will be following
> a! = 720 = 1*5^4 + 0*5^3 + 3*5^2 + 4*5^1 + 0
> and 5^4<=720<5^5 and from these reason 4 is a(6) in A127032
> In pentanary positional system 6! is number 10340
>
> BEST WISHES
> ARTUR
>
>
> Dnia 05-01-2007 o 09:40:45 Max A. <maxale at gmail.com> napisał(a):
>
>> Artur and SeqFans,
>>
>> I've checked the book "The Little Book of Big Primes" (2nd ed.) by
>> Ribenboim, Section 2.D at page 24. It indeed states the following:
>>
>> "In 1808, Legendre the exact power p^m of the prime p that divides a
>> factorial a! (so that p^(m+1) does not divide a!)."
>>
>> which seems to be consistent with the comment
>>
>> %C A125552 A. M. Legendre in 1808 gave a formula for finding m.
>>
>> and similar comments in other sequences (A127032, A127033, A127034,
>> A127035, A127036, A127037, A127039). There is a problem, though.
>>
>> The Legendre's formula is irrelevant to these sequences, since it
>> indeed determines the maximum power of p *dividing* n! (as Ribenboim
>> said) while Artur's sequences define the maximum power of p *not
>> exceeding* n!.
>> So the comments like the one quoted above should be removed from these
>> sequences.
>>
>> P.S. As I see, Neil has already corrected these sequences according to
>> my previous message. That's fine but somehow "less or equal" symbol in
>> the sequence names got incorrectly replaced with strict < symbol.
>> Neil, please fix it up as well.
>>
>> Thanks,
>> Max
>>
>> On 1/4/07, Artur <grafix at csl.pl> wrote:
>>> Dear Max,
>>> I was take these title from Ribenboim book 2.2.4
>>> ARTUR
>>>
>>> Dnia 04-01-2007 o 22:08:11 Max A. <maxale at gmail.com> napisał(a):
>>>
>>> > On 1/4/07, N. J. A. Sloane <njas at research.att.com> wrote:
>>> >
>>> >> Also, just to make things more difficult, Artur submitted several
>>> pairs
>>> >> of sequences with the same A-numbers, for example:
>>> >> %I A127030
>>> >> %S A127030 0, 1, 2, 4, 5, 7, 9, 11, 13, 15, 18, 20, 22
>>> >> %N A127030 Maximal value of power m such that 3^m divided n! (3^m
>>> <
>>> >> n!)
>>> >
>>> > I think the name of this and other similar sequences (A127032,
>>> > A127033, A127034, A127035, A127036, A127037, A127039) is confusing.
>>> It
>>> > has nothing to do with the division (but rather with taking a
>>> > logarithm), and it should be renamed to
>>> > "Maximal value of power m such that 3^m < n!"
>>> > with a formula
>>> > a(n) = floor( log(n!) / log(3) ).
>>> >
>>> > Max
>>>
>>>
>>>
>
>
>
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