A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2

Offset

1,1

Comments

The GCD of integers x^(n+1)-x, for all integers x. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 19 2002

If each x in a ring satisfies x^(n+1)=x, the characteristic of the ring is a divisor of a(n) (Rosenblum 1977). - Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008

The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. To add a(0) = 1 has been proposed in A141056. - Peter Luschny, Apr 29 2009

For N > 1, a(n) is the greatest number k such that x*y^n ==y*x^n (mod k) for any integers x and y. Example: a(19) = 798 because x*y^19 ==y*x^19 (mod 798). - Michel Lagneau, Apr 21 2012

a(n) is the largest k such that b^(n+1) == b (mod k) for every integer b. - Mateusz Szymański, Feb 18 2016, corrected by Thomas Ordowski, Jul 01 2018

When n is even, a(n) is the product of the distinct primes dividing the denominator of zeta(1-n), where zeta(s) is the Riemann zeta function. - Griffin N. Macris, Jun 13 2016

If n+1 is prime, then A002322(a(n)) = n. Composite numbers n+1 such that A002322(a(n)) = n are in A317210. - Max Alekseyev and Thomas Ordowski, Jul 09 2018

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351-352. [Peter Luschny, Apr 29 2009]

S. C. Locke and A. Mandel, Problem E 2901, American Mathematical Monthly 88 (1981), p. 538. Solution in Vol. 90 (1983), pp. 212-213. [Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Jul 31 2008]

D. M. Rosenblum, Problem 1019, Mathematics Magazine 50 (1977), p. 164. Solution by T. Orloff in Vol. 52 (1979), p. 50.

Wikipedia, Bernoulli number

Formula

a(2*k) = A091137(2*k)/A091137(2*k-1). - Paul Curtz, Aug 05 2008

a(n) = product_{p prime, p-1 divides n}. - Eric M. Schmidt, Aug 01 2013

a(2n-1) = 2. - Robert G. Wilson v, Jul 23 2018

Example

1/2, 5/6, 1/2, 31/30, 1/2, 41/42, 1/2, 31/30, 1/2, 61/66, 1/2, 3421/2730, 1/2, 5/6, 1/2, 557/510, ...

Maple

A027760 := proc(n) local s, p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end: # R. J. Mathar, Aug 12 2008

Mathematica

clausen[n_] := Product[i, {i, Select[ Map[ # + 1 &, Divisors[n]], PrimeQ]}]
Table[clausen[i], {i, 1, 20}] (* Peter Luschny, Apr 29 2009 *)
f[n_] := Times @@ Select[Divisors@n + 1, PrimeQ]; Array[f, 56] (* Robert G. Wilson v, Apr 25 2012 *)

Prog

(PARI) a(n)=denominator(sumdiv(n, d, if(isprime(d+1), 1/(d+1)))) \\ Charles R Greathouse IV, Jul 08 2011
(PARI) a(n)=my(pr=1); fordiv(n, d, if(isprime(d+1), pr*=d+1)); pr \\ Charles R Greathouse IV, Jul 08 2011
(Sage)
def A027760(n):
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+1, divisors(n))))
[A027760(n) for n in (1..56)]  # Peter Luschny, May 23 2013

Crossrefs

Cf. A002997, A027759, A027642, A141056, A317210.

Sequence in context: A131980 A217448 A280705 * A141056 A141498 A284004

Adjacent sequences: A027757 A027758 A027759 * A027761 A027762 A027763

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

Formula submitted with A141417 added by R. J. Mathar, Nov 17 2010

Status

approved