A242194 Least prime divisor of E_{2*n} which does not divide any E_{2*k} with k < n, or 1 if such a primitive prime divisor of E_{2*n} does not exist, where E_m denotes the m-th Euler number given by A122045.

1, 5, 61, 277, 19, 13, 47, 17, 79, 41737, 31, 2137, 67, 29, 15669721, 930157, 4153, 37, 23489580527043108252017828576198947741, 41, 137, 587, 285528427091, 5516994249383296071214195242422482492286460673697, 5639, 53, 2749, 5303, 1459879476771247347961031445001033, 6821509

Offset

1,2

Comments

Conjecture: a(n) is prime for any n > 1.

It is known that (-1)^n*E_{2*n} > 0 for all n = 0, 1, ....

See also A242193 for a similar conjecture involving Bernoulli numbers.

Links

Peter Luschny, Table of n, a(n) for n = 1..82, (a(1)..a(34) from Zhi-Wei Sun, a(35)..a(38) from Jean-François Alcover).

Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Example

a(4) = 277 since E_8 = 5*277 with 277 not dividing E_2*E_4*E_6, but 5 divides E_4 = 5.

Mathematica

e[n_]:=Abs[EulerE[2n]]
f[n_]:=FactorInteger[e[n]]
p[n_]:=p[n]=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[e[n]<2, Goto[cc]]; Do[Do[If[Mod[e[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, 30}]
(* Second program: *)
LPDtransform[n_, fun_] := Module[{}, d[p_, m_] := d[p, m] = AllTrue[ Range[m-1], ! Divisible[fun[#], p]&]; f[m_] := f[m] = FactorInteger[ fun[m]][[All, 1]]; SelectFirst[f[n], d[#, n]&] /. Missing[_] -> 1];
a[n_] := a[n] = LPDtransform[n, Function[k, Abs[EulerE[2k]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 38}]  (* Jean-François Alcover, Jul 28 2019, non-optimized adaptation of Peter Luschny's Sage code *)

Prog

(Sage) # uses[LPDtransform from A242193]
A242194list = lambda sup: [LPDtransform(n, lambda k: euler_number(2*k)) for n in (1..sup)]
print(A242194list(16)) # Peter Luschny, Jul 26 2019

Crossrefs

Cf. A000040, A000364, A122045, A242169, A242170, A242171, A242173, A242193, A242195.

Sequence in context: A142643 A201848 A087871 * A046976 A092838 A196296

Adjacent sequences: A242191 A242192 A242193 * A242195 A242196 A242197

Keyword

hard,nonn

Author

Zhi-Wei Sun, May 07 2014

Status

approved