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A027760 Denominator of sum_{p prime, p-1 divides n} 1/p. 25
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 (list; graph; refs; listen; history; text; internal format)
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]

REFERENCES

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.

LINKS

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

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

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. A027759, A027642, A141056.

Sequence in context: A076743 A131980 A217448 * A141056 A141498 A225481

Adjacent sequences:  A027757 A027758 A027759 * A027761 A027762 A027763

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Added formula submitted with A141417 - R. J. Mathar, Nov 17 2010

STATUS

approved

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Last modified October 2 12:28 EDT 2014. Contains 247541 sequences.