[seqfan] Re: A conjecture about the numerator of Euler(2n-1,n)

[Vladimir Shevelev]

(Please, ignore the previous text)

Dear Seq Fans,

In new sequence A292911 (Numbers n such that 
A291897(n) is divisible by (2*n-1)^3) I did the second
conjecture that for n>=2 the numbers {2*a(n)-1} are
consecutive primes of the form 4*k+1 (cf. A002144).
But by n=553 Peter Moses disproved this conjecture.
2*553-1 is the smallest number which is the product
of three distinct (4k+1)-primes.  
But I continue to believe that for every (4k+1)-prime
p, (p+1)/2 is a number n such that A291897(n) is divisible by 
(2*n-1)^3), i.e., {2A292911(n)-1} contains all such primes
[corrected 2-nd conjecture]
(maybe, it contains products of odd number of
distinct such primes).
It still seems to me that to prove this 2-nd conjecture
is a very difficult problem. However, I think that it
is highly interesting since does more precise
the conjecture in A291897. 

Best regards,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 22 September 2017 23:06
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: A conjecture about the numerator of Euler(2n-1,n)

Yes, Peter Luschny simplified one of my
formula in A291897 (see there), using
a property of A002425, so you are
right. Thank you very much for the
verification of my conjecture up to
5000.

Best regards,
Vladimir


________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of rgwv at rgwv.com [rgwv at rgwv.com]
Sent: 22 September 2017 20:56
To: 'Sequence Fanatics Discussion list'
Subject: [seqfan] Re: A conjecture about the numerator of Euler(2n-1,n)

Are the log2 of the denominators of Euler(2n-1,n) the sequence https://oeis.org/A001511: The ruler function?

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Vladimir Shevelev
Sent: Friday, 22 September, 2017 9:38 AM
To: seqfan at list.seqfan.eu
Subject: [seqfan] A conjecture about the numerator of Euler(2n-1,n)

Dear SeqFans,

Now I have just submitted a new sequence
A291897 "Numerator of Euler(2n-1,n)".
I proved that
a(n)=2(-1)^n*A006519(2n)*(1^(2n-1) - 2^(2n-1)+
...+(-1)^n*(n-1)^(2n-1)) + A002425(n).
The sequence begins 1, 9, 125, 32977,...
My observation is about an interesting property:
a(n) is divisible by (2*n-1)^2 (conjecture).
Note that sometimes, maybe, for prime n of the form 4*k+1, a(n) is divisible by (2n-1)^3 ( for example, for  n=1,5,13,17,...).
Can anyone analyze this property?

Best regards,
Vladimir

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