OFFSET
1,5
COMMENTS
Conjecture: a(n) is prime for any n > 4.
It is known that (-1)^(n-1)*B_{2*n} > 0 for all n > 0.
See also A242194 for a similar conjecture involving Euler numbers.
LINKS
Peter Luschny, Table of n, a(n) for n = 1..103, (a(1)..a(60) from Zhi-Wei Sun)
Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
EXAMPLE
a(14) = 9349 since the numerator of |B_{28}| is 7*9349*362903 with B_2*B_4*B_6*...*B_{26} not congruent to 0 modulo 9349, but B_{14} == 0 (mod 7).
MATHEMATICA
b[n_]:=Numerator[Abs[BernoulliB[2n]]]
f[n_]:=FactorInteger[b[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[b[n]<2, Goto[cc]]; Do[Do[If[Mod[b[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}]; Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}]; Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 35}]
(* Second program: *)
LPDtransform[n_, fun_] := Module[{d}, d[p_] := AllTrue[Range[n - 1], !Divisible[fun[#], p]&]; SelectFirst[FactorInteger[fun[n]][[All, 1]], d] /. Missing[_] -> 1];
A242193list[sup_] := Table[LPDtransform[n, Function[k, Abs[BernoulliB[2k]] // Numerator]], {n, 1, sup}]
A242193list[35] (* Jean-François Alcover, Jul 27 2019, after Peter Luschny *)
PROG
(Sage)
def LPDtransform(n, fun):
d = lambda p: all(not p.divides(fun(k)) for k in (1..n-1))
return next((p for p in prime_divisors(fun(n)) if d(p)), 1)
A242193list = lambda sup: [LPDtransform(n, lambda k: abs(bernoulli(2*k)).numerator()) for n in (1..sup)]
print(A242193list(35)) # Peter Luschny, Jul 26 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 07 2014
STATUS
approved