A166062 a(n) = denominator(Bernoulli(prime(n) - 1)).

2, 6, 30, 42, 66, 2730, 510, 798, 138, 870, 14322, 1919190, 13530, 1806, 282, 1590, 354, 56786730, 64722, 4686, 140100870, 3318, 498, 61410, 4501770, 33330, 4326, 642, 209191710, 1671270, 4357878, 8646, 4110, 274386, 4470, 2162622, 1794590070, 130074

Offset

1,1

Comments

Divisibility through terms of A008578 is a consequence of the Staudt-Clausen theorem.

(Vaguely similar divisibility properties are considered in A165248 and A165943.)

The first 250 entries are all different. Is this true in general?

Would sorting the entries yield the full A090801?

a(n) > 1 is the largest number k such that x*y^p == y*x^p (mod k) for all integers x and y, where p = prime(n). Example: x*y^19 == y*x^19 (mod 798). - Michel Lagneau, Apr 19 2012

Comment from Herbert Kociemba, May 29 2020: (Start)

For each n there is exactly one member of the sequence whose factorization has prime(n) as its largest prime factor, namely a(n). From this we conclude:

1. All elements of the sequence are different.

2. Not all denominators of Bernoulli numbers appear in this sequence. For example the denominator of B(20), 330=2*3*5*11 never appears because the unique sequence element with largest prime divisor 11=prime(5) is a(5)=2*3*11. (End)

Links

Peter Luschny, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem.

Formula

a(n) = A027642(A008578(n) - 1).

Maple

seq(denom(bernoulli(ithprime(n)-1)), n=1..38); # Peter Luschny, Jul 14 2019

Mathematica

Table[Denominator[BernoulliB[n - 1]], {n, Prime[Range[38]]}] (* Harvey P. Dale, Apr 22 2012 *)
Table[GCD @@ Table[(n^k - n), {n, 2, 13}], {k, Prime[Range[100]]}] (* Increase n to 80 and k to 1000 for first thousand terms. - Herbert Kociemba, May 05 2020 *)
a[i_] := Times @@ Select[Prime[Range[i]], Mod[Prime[i] - 1, # - 1] == 0&]; Table[a[i], {i, 1, 100}](* Herbert Kociemba, May 06 2020 *)

Prog

(PARI) a(n)=denominator(bernfrac(prime(n)-1)) \\ Charles R Greathouse IV, Apr 30 2012

Crossrefs

Cf. A071772, A110936.

Sequence in context: A286652 A265501 A090801 * A100194 A229882 A325986

Adjacent sequences: A166059 A166060 A166061 * A166063 A166064 A166065

Keyword

nonn

Author

Paul Curtz, Oct 05 2009

Extensions

Edited by Peter Luschny, Jul 14 2019

Status

approved