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