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%I #47 Mar 27 2020 06:58:44
%S 1,5,61,277,19,13,47,17,79,41737,31,2137,67,29,15669721,930157,4153,
%T 37,23489580527043108252017828576198947741,41,137,587,285528427091,
%U 5516994249383296071214195242422482492286460673697,5639,53,2749,5303,1459879476771247347961031445001033,6821509
%N 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.
%C Conjecture: a(n) is prime for any n > 1.
%C It is known that (-1)^n*E_{2*n} > 0 for all n = 0, 1, ....
%C See also A242193 for a similar conjecture involving Bernoulli numbers.
%H Peter Luschny, <a href="/A242194/b242194.txt">Table of n, a(n) for n = 1..82</a>, (a(1)..a(34) from Zhi-Wei Sun, a(35)..a(38) from Jean-François Alcover).
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv preprint arXiv:1405.0290 [math.NT], 2014.
%e a(4) = 277 since E_8 = 5*277 with 277 not dividing E_2*E_4*E_6, but 5 divides E_4 = 5.
%t e[n_]:=Abs[EulerE[2n]]
%t f[n_]:=FactorInteger[e[n]]
%t p[n_]:=p[n]=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
%t 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}]
%t (* Second program: *)
%t 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];
%t a[n_] := a[n] = LPDtransform[n, Function[k, Abs[EulerE[2k]]]];
%t 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 *)
%o (Sage) # uses[LPDtransform from A242193]
%o A242194list = lambda sup: [LPDtransform(n, lambda k: euler_number(2*k)) for n in (1..sup)]
%o print(A242194list(16)) # _Peter Luschny_, Jul 26 2019
%Y Cf. A000040, A000364, A122045, A242169, A242170, A242171, A242173, A242193, A242195.
%K hard,nonn
%O 1,2
%A _Zhi-Wei Sun_, May 07 2014
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