A242193 Least prime p such that B_{2*n} == 0 (mod p) but there is no k < n with B_{2k} == 0 (mod p), or 1 if such a prime p does not exist, where B_m denotes the m-th Bernoulli number.

1, 1, 1, 1, 5, 691, 7, 3617, 43867, 283, 11, 103, 13, 9349, 1721, 37, 17, 26315271553053477373, 19, 137616929, 1520097643918070802691, 59, 23, 653, 417202699, 577, 39409, 113161, 29, 2003, 31, 1226592271, 839, 101, 688531

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

Cf. A000040, A027641, A242169, A242170, A242171, A242173, A242194, A242195.

Sequence in context: A198597 A180315 A332300 * A326727 A090947 A176840

Adjacent sequences: A242190 A242191 A242192 * A242194 A242195 A242196

Keyword

nonn

Author

Zhi-Wei Sun, May 07 2014

Status

approved