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A242195 Least prime divisor of the n-th tangent number T_n which does not divide any T_k with k < n, or 1 if such a primitive prime divisor of T_n does not exist. 7
1, 2, 1, 17, 31, 691, 43, 257, 73, 41, 89, 103, 2731, 113, 151, 37, 43691, 109, 174763, 61681, 337, 59, 178481, 97, 251, 157, 39409, 113161, 67, 1321, 266689, 641, 839, 101, 281, 433, 223, 229, 121369, 631 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: (i) a(n) is prime for any n > 3.

(ii) For the n-th Springer number S_n given by A001586, if n is greater than one and not equal to 5, then S_n has a prime divisor which does not divide any S_k with k < n.

See also A242193 and A242194 for similar conjectures involving Bernoulli numbers and Euler numbers.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..54

EXAMPLE

a(4) = 17 since T_4 = 2^4*17 with 17 dividing none of T_1 = 1, T_2 = 2 and T_3 = 2^4.

MATHEMATICA

t[n_]:=(-1)^(n-1)*2^(2n)(2^(2n)-1)BernoulliB[2n]/(2n)

f[n_]:=FactorInteger[t[n]]

p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]

Do[If[t[n]<2, Goto[cc]]; Do[Do[If[Mod[t[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, 40}]

PROG

(Sage) # uses[LPDtransform from A242193]

def Tnum(n): return (-1)^(n-1)*2^(2*n)*(2^(2*n)-1)*bernoulli(2*n)/(2*n)

A242195list = lambda sup: [LPDtransform(n, Tnum) for n in (1..sup)]

print(A242195list(40)) # Peter Luschny, Jul 26 2019

CROSSREFS

Cf. A000040, A000182, A001586, A242169, A242170, A242171, A242173, A242193, A242194.

Sequence in context: A266827 A316226 A160468 * A012889 A013072 A012895

Adjacent sequences:  A242192 A242193 A242194 * A242196 A242197 A242198

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 07 2014

STATUS

approved

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Last modified July 12 13:21 EDT 2020. Contains 335663 sequences. (Running on oeis4.)