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.

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

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: A316226 A352138 A160468 * A012889 A013072 A012895

Adjacent sequences: A242192 A242193 A242194 * A242196 A242197 A242198

Keyword

nonn

Author

Zhi-Wei Sun, May 07 2014

Status

approved