[seqfan] Re: Comment on A151750 [erratum]
David Wilson
davidwwilson at comcast.net
Sat Aug 1 23:33:10 CEST 2009
I have to rescind my assertion that
b does not divide choose(2n, n) <=> all b-ary digits of n are < b/2.
for odd b. It is not true for prime b, but not true for 9, and I assume for
any composite. However, I can prove it b = p prime.
Theorem: For prime p
p does not divide choose(2n, n) <=> all p-ary digits of n are < p/2.
Proof: The exponent of prime p in n! is given by
[1] e_p(n) = (n - s_p(n))/(p -1)
where s_p(n) is the sum of the p-ary digits of n (well known to those who
well know it).
So p does not divide choose(2n, n) = (2n)! / (n!)^2 precisely when
e_p(2n) - 2e_p(n) = 0
Together with [1], this implies
s_p(2n) = 2 s_p(n)
But the sum of p-ary digits in 2n is twice the sum of the p-ary digits in n
precisely when there is no carry in the sum n+n, that is to say, every p-ary
digit of n is < p/2, QED.
----- Original Message -----
From: "David Wilson" <davidwwilson at comcast.net>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Saturday, August 01, 2009 4:44 PM
Subject: [seqfan] Re: Comment on A151750 [erratum]
Sorry, I got a sign wrong.
----- Original Message -----
From: "David Wilson" <davidwwilson at comcast.net>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Saturday, August 01, 2009 4:28 PM
Subject: [seqfan] Re: Comment on A151750
For b >= 2, let
S_b = { n : b does not divide choose(2n, n) }
I believe you can prove
S_b = { n : every digit in b-ary representation of n is <= (b-1)/2 }
This would imply that
|{ n in S_b : n < b^k }| = ((b-1)/2)^k
which would in turn give us a density estimate for S_b.
We can then compute a density estimate S_3 intersect S_5 intersect S_7
intersect S_11 and decide whether we expect this set to be infinite or
finite.
----- Original Message -----
From: "peter.luschny" <peter.luschny at googlemail.com>
To: <seqfan at list.seqfan.eu>
Sent: Friday, July 31, 2009 3:44 PM
Subject: [seqfan] Comment on A151750
Inspired by Niel's yesterdays submission A151750
I tried to integrate this difficult problem in the
framework of divisibility properties of the swinging
factorial. (See also
http://www.luschny.de/math/primes/SwingingPrimes.html
---
Let S be a sequence and p a prime number.
S(n) is 'p-primefree' iff no prime <= p divides S(n).
Now we can associate to S
pf(S,i) -> seq(n, such that S(n) is ith_prime-primefree).
For example assume S to be the *odd kernel* of the
swinging factorial $ (A056040).
Then 6$ is 3-primefree, 20$ is 5-primefree, 1512$
is 7-primefree, 6320$ is 11-primefree, ...
The first few values of the associated sequences are:
pf(odd($),1) = 1,2,6,7,8,18,19,20,24,25,26,54,..
pf(odd($),2) = 1,2,20,24,54,60,61,62,72,73,74,504..
pf(odd($),3) = 1,2,20,1512,1513,1514,..
pf(odd($),4) = 1,2,6320,...
pf(odd($),5) = 1,2,...
Which of these sequences are finite? And if they are
finite, what is the largest n such that S(n) is in PF(S,i)?
i=1: infinite (Lucas),
i=2: infinite (Erdös et al.),
i=3: infinite conjectured (?),
i=4: finite conjectured, max=6320,
i>=5: finite? infinite?, max=2 always?
See the comments in A030979 and A151750.
The 3160 from A151750 and of A129489 is 6320 / 2.
A129508, doubled, is a subsequence of pf(odd($),2).
A030979, doubled, is a subsequence of pf(odd($),3).
See also A129488.
A possible attempt to concentrate these relations
in a sequence:
pf(odd($)): i -> min {pf(odd($),i) > primorial(i)}.
6,20,1512,6320,?,?,.. ( ? = 2 )
======================
Many questions open. Any comment appreciated.
Cheers Peter
--------------------------------------------------
swing := proc(n) option remember;
if n = 0 then 1 elif irem(n, 2) = 1 then
swing(n-1)*n else 4*swing(n-1)/n fi end:
oddprimorial := proc(n) option remember; local i;
mul(ithprime(i+1),i=1..n) end:
pf := proc(n,m) local v,k,p; v := 0;
p := oddprimorial(n);
for k from 1 to m do
if igcd(swing(k),p) = 1 then v := k fi;
if v > p then break fi od; [n,v] end:
submit := proc(n) pf(n,10000) end;
seq(submit(i),i=1..10);
[1, 6], [2, 20], [3, 1512], [4, 6320], [5, 2],
[6, 2], [7, 2], [8, 2], [9, 2], [10, 2], ...
--------------------------------------------------
_______________________________________________
Seqfan Mailing list - http://list.seqfan.eu/
_______________________________________________
Seqfan Mailing list - http://list.seqfan.eu/
_______________________________________________
Seqfan Mailing list - http://list.seqfan.eu/
More information about the SeqFan
mailing list