A241601 - OEIS

A241601

Largest divisor of A246006(n) whose prime factors are all >= n+2.

1, 1, 1, 1, 1, 1, 1, 61, 1, 277, 1, 50521, 691, 41581, 1, 199360981, 3617, 228135437, 43867, 2404879675441, 174611, 14814847529501, 77683, 69348874393137901, 236364091, 238685140977801337, 657931, 4087072509293123892361, 3392780147, 454540704683713199807

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Notice: Not all a(n) are 1 or primes, the first example is a(11) = 50521, it equals 19*2659.

a(2n) is a product of powers of Bernoulli irregular primes (A000928), with the exception of n = 0,1,2,3,4,5,7.

a(2n+1) is a product of powers of Euler irregular primes (A120337), with the exception of n = 0,1,2.

Conjectures: All terms are squarefree, and there are infinitely many n such that a(n) is prime.

a(n) = 1 iff n is in the set {0, 1, 2, 3, 4, 5, 6, 8, 10, 14}.

a(n) is prime for n = {7, 9, 12, 16, 17, 18, 26, 34, 36, 38, 39, 42, 49, 74, 114, 118, ...}.

All prime factors of a(n) are irregular primes (Bernoulli or Euler) and with an irregular pair to n: (61, 7), (277, 9), (19, 11), (2659, 11), (691, 12), (43, 13), (967, 13), (47, 15), (4241723, 15), (3617, 16), (228135437, 17), (43867, 18), (79, 19), (349, 19), (84224971, 19), ...

Number of ns such that a prime p divides a(n) is the irregular index of p, for example, 67 divides both a(27) and a(58), so it has irregular index two.

a(149) is the first a(n) which is not completely factored (with a 202-digit composite remaining).

LINKS

Table of n, a(n) for n=0..29.

FORMULA

a(2n) = |A001067(n)| = |A120082(2n)| = |A141590(n)| = |A060054(n)|.

a(2n+1) = A091912(n).

MATHEMATICA

b[n_] := Numerator[BernoulliB[2 n]/(2 n)];

c[n_] := Numerator[SeriesCoefficient[Log[Tan[x]+1/Cos[x]], {x, 0, 2n+1}]];

a[0] = 1; a[n_] := If[EvenQ[n], b[n/2] // Abs, c[(n-1)/2]];

Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jul 03 2019 *)

CROSSREFS