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
KEYWORD
hard,nonn
AUTHOR
Zhi-Wei Sun, May 07 2014
STATUS
approved