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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A203925 A198597 A180315 * A326727 A090947 A176840

Adjacent sequences:  A242190 A242191 A242192 * A242194 A242195 A242196

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 07 2014

STATUS

approved

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Last modified December 10 15:09 EST 2019. Contains 329896 sequences. (Running on oeis4.)