A241601 - OEIS

%I #74 Jul 03 2019 06:14:20

%S 1,1,1,1,1,1,1,61,1,277,1,50521,691,41581,1,199360981,3617,228135437,

%T 43867,2404879675441,174611,14814847529501,77683,69348874393137901,

%U 236364091,238685140977801337,657931,4087072509293123892361,3392780147,454540704683713199807

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

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

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

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

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

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

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

%C 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), ...

%C 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.

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

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

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

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

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

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

%t Table[a[n], {n, 0, 29}] (* _Jean-François Alcover_, Jul 03 2019 *)

%Y Cf. A246006.

%Y Cf. A000928, A120337.

%Y Cf. A001067, A120082, A141590, A060054, A091912.

%K nonn

%O 0,8

%A _Eric Chen_, Dec 15 2014