Bernoulli/Genocchi numbers

[benoit]

This is probably not directly related to previous interesting  
discussion. I made last year some  "light" experimental investigations  
(need to be confirmed) related to congruence properties of Genocchi  
numbers of first kind  : G(k)=2*(4^k-1)*B(2k)=-A001469(k) and more  
generally of numbers G(k,m)= r(m)*(m^(2k)-1)*B(2k) where r(m) is the  
smallest rational value making G(k,m) integer for all k (cf. A089655).

(i) (G(k) mod n) k>0  appears to become a periodic sequence with period  
length L(n)  satisfying for odd n=2m+1 :

L(n):= n*prod(p | n, (p-1)/2)=n/(-2)^omega(n)*sum(d | n, d*mu(d))

Hence for p odd prime : L(p)=p*(p-1)/2.  Something to do with Roland  
Bacher's conjecture ?

"Conjecture: all indices yielding a given prime p form an arithmetic  
progression of step ((p-1)/2)*p."

letting : a(n)= (2n+1)*prod(p | 2n+1, (p-1)/2) n>=0, we get a sequence  
not in OEIS :

1,3,10,21,9,55,78,30,136,171,63,253,50,27,406,465,165,210,666,234,820,90 
3,90,1081,
147,408,1378,550,513,1711,1830,189,780,2211,759,2485,2628,150,1155,3081, 
81,3403,
1360,1218,3916,1638,1395,1710,4656,495,5050,

Possible Name : "conjectured" values of period length of sequence of  
Genocchi numbers of first kind modulo 2n+1.

also : a(n)= n*prod(p | n, (p-1)/2) for n>=1 is not in OEIS too :

1,1,3,2,10,3,21,4,9,10,55,6,78,21,30,8,136,9,171,20,63,55,253,12,50,78,2 
7,42,406,30,465,16,
165,136,210,18,666,171,234,40,820,63,903,110,90,253,1081,24,147,50,408,1 
56,1378,27,550,
84,513,406,1711,60,1830,465,189,32,780,165,2211,272,759,210

(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)

I noticed :  {G(2k+1)/(2k+1) mod 2^n} k>0  has period length 2^(n-1)
And :
if p is prime==1 mod4 :  {G(2k+1)/(2k+1) mod p^n} k>0  has period  
length p^(n-1)*(p-1)/4
if p is prime==3 mod4 :  {G(2k+1)/(2k+1) mod p^n} k>0  has period  
length p^(n-1)*(p-1)/2

Therefore one should expect for odd n something like :

{G(2k+1)/(2k+1) mod n} k>0  has period length L(n):= prod(  p==3 mod4 |  
n, p^(n-1)*(p-1)/2)* prod(  p==1 mod4 | n, p^(n-1)*(p-1)/4)

(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)

(iv) etc.

BC
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