Bernoulli/Genocchi numbers

[Ralf Stephan]

Benoit Cloitre:
> letting : a(n)= (2n+1)*prod(p | 2n+1, (p-1)/2) n>=0, we get a sequence 
> not in OEIS : 
> 
> 1,3,10,21,9,55,78,30,136,171,63,253,50,27,406,465,165,210,666,234,820,903,90,1081, 

which seems to be 1/2 the values of the odd bisection of A088659,
i.e. (2n+1)/2 * (p-1) with p the largest prime factor of 2n+1...


> (ii) Since (2k+1) divides G(2k+1) for any k>=0 I considered the integer 
> sequence {G(2k+1) /(2k+1)} k>=0 .  
> That's sequence : 
> 1,31,5461,3202291,4722116521,14717667114151...(A090681) 

which seems related to A012670, A002425, A089171, A012853 


> (iii) I also considered H(k)= 2^valuation(2*k,2) * G(2*k)/(2*k) which 
> is an integer sequence (=-A002425). Then {H(k) mod3} k>0  is 
> interestingly related to the first Feigenbaum symbolic sequence. Namely 
>  :  H(k) mod3 = 1+A035263(k)   

this has nothing to do with the Genocchi numbers, and holds for
2^valuation(2n,2) as well!


ralf