cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

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

Examples

			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, ...

Links

Crossrefs

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

Programs

  • 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 *)
  • 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

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

Extensions

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

A248614 Rank of the n-th distinct value of the Bernoulli denominators in the sequence of the denominators of the Bernoulli numbers.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 12, 16, 18, 20, 22, 28, 30, 36, 40, 42, 44, 46, 48, 52, 58, 60, 66, 70, 72, 78, 80, 82, 84, 88, 90, 92, 96, 100, 102, 106, 108, 110, 112, 116, 120, 126, 130, 132, 136, 138, 140, 144, 148, 150, 156, 162, 164, 166, 172, 174, 176, 178, 180, 190, 192
Offset: 0

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Author

Paul Curtz, Oct 09 2014

Keywords

Comments

Consider sequence A027642 of the denominators of the Bernoulli numbers and the reduced sequence b(n) = 1, 2, 6, 30, 42, 66,... if duplicates are removed (which is 1, 2 followed by A090126). a(n) shows the smallest index --place of first appearance-- of b(n) in the full list A027642.
If n is of the form A002322(p*q) with p*q semiprime, then n is a term. The number 3652 is a term, but it is not of the form A002322(p*q), as Carl Pomerance noted. - Thomas Ordowski, Apr 28 2021; in place of an incorrect comment by Filip Zaludek, Sep 23 2016
For n > 0, numbers n such that A002322(A027642(n)) = n. - Thomas Ordowski, Jul 11 2018
Carl Pomerance (in answer to my question) proved that the set of these numbers has asymptotic density zero. - Thomas Ordowski, Apr 28 2021

Examples

			b(2)=6 appears first in A027642(2), so a(2)=2. b(4)=42 appears first as A027642(6)=42, so a(4)=6. b(5)=66 appears first as A027642(10), so a(5)=10.

Crossrefs

Cf. A002322, A002445, A027642, A090126, A090801, A317210.

Programs

  • Mathematica
    BB = Table[Denominator[BernoulliB[n]], {n, 2, 400, 2}]; For[t = BB; n = 1, n <= Length[t], n++, p = Position[t, t[[n]]] // Rest; t = Delete[t, p]]; reducedBB = Join[{1, 2}, t]; a[0] = 0; a[1] = 1; a[n_] := 2*Position[BB, reducedBB[[n+1]], 1, 1][[1, 1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 16 2014 *)
  • PARI
    L=List(); N=60; forprime(p=2, N*N, forprime(q=p, N*N, listput(L, lcm(p-1,q-1)) )); listsort(L, 1); for (i=1, N, print1(L[i], ", ")) \\ Filip Zaludek, Sep 23 2016

A346468 a(n) = (n-1) / A346467(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 1, 7, 2, 9, 1, 11, 1, 13, 7, 15, 1, 17, 1, 19, 1, 21, 1, 23, 2, 25, 13, 27, 1, 29, 1, 31, 2, 33, 17, 35, 1, 37, 19, 39, 1, 41, 1, 43, 1, 45, 1, 47, 1, 49, 5, 51, 1, 53, 3, 55, 2, 57, 1, 59, 1, 61, 31, 63, 4, 65, 1, 67, 17, 69, 1, 71, 1, 73, 37, 75, 19, 77, 1, 79, 1, 81, 1, 83, 1, 85, 43, 87, 1, 89
Offset: 1

Views

Author

Antti Karttunen and Thomas Ordowski, Jul 22 2021

Keywords

Comments

Numbers n such that a(n) = 1 are A248614(m)+1 for m > 0. These are all primes together with A317210. The set of these numbers has zero asymptotic density.

Links

Crossrefs

Cf. A002322, A027642, A248614, A317210, A346467.

Programs

  • Mathematica
    {0}~Join~Array[#/CarmichaelLambda@ Denominator@ BernoulliB@ # &, 89] (* Michael De Vlieger, Nov 23 2021 *)
  • PARI
    A346468(n) = if(1==n,0,my(m=1); fordiv(n-1,d,if(isprime(1+d),m = lcm(m,d))); ((n-1)/m));

Formula

a(n) = (n-1) / A346467(n).
a(n) = (n-1) / A002322(A027642(n-1)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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