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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. 8
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 (list; graph; refs; listen; history; text; internal format)
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) # Function LPDtransform is defined in 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

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Last modified December 14 19:27 EST 2019. Contains 329987 sequences. (Running on oeis4.)