A106741 Numbers n such that n divides the denominator of 2n-th Bernoulli number.

1, 2, 3, 6, 10, 21, 30, 42, 78, 110, 210, 330, 390, 546, 903, 930, 1218, 1806, 1830, 2310, 2530, 2730, 4134, 4290, 6090, 6162, 6510, 7590, 9030, 10230, 12090, 12246, 12810, 14910, 15834, 20130, 20670, 22110, 23478, 23790, 28938, 30030, 30810, 43134

Offset: 1

Author

Benoit Cloitre, May 15 2005

Keywords

• nonn

Comments

Numbers n such that the congruence k^(2n+1) == k (mod n) is true for 1<=k<=n. - Michel Lagneau, May 02 2012

In 2005, B. C. Kellner proved E. W. Weisstein's conjecture that denom(B_n) = n only if n = 1806. - Jonathan Sondow, Oct 14 2013.

Links

• Joerg Arndt, Table of n, a(n) for n = 1..594 (terms <= 10^8)
• Bernd C. Kellner, The equation denom(B_n) = n has only one solution, preprint 2005.
• Victor Miller, Re: Q about a property of Bernoulli denominators, NMBRTHRY list, May 5, 2012
• Eric Weisstein's World of Mathematics, Bernoulli Number

Crossrefs

Cf. A002445, A014117.

Programs

Maple

for n from 1 to 10000 do:
    m:=2*n+1: i:=1:
    for k from 1 to n while(k &^ m mod n =k) do: i:=i+1: od:
    if i=n then print(n) fi:
od: # Michel Lagneau, May 02 2012
A106741_list := proc(searchlimit) local isA106741, i;
isA106741 := proc(n)
  numtheory[divisors](2*n);
  map(i->i+1,%);
  select(isprime,%);
  mul(i,i=%) mod n = 0;
  if % then n else NULL fi end:
seq(isA106741(i),i=1..searchlimit) end:
A106741_list(30000); # Peter Luschny, May 04 2012

Mathematica

okQ[n_] := AllTrue[Range[n], PowerMod[#, 2n+1, n] == Mod[#, n]&];
Reap[For[n = 1, n < 50000, n++, If[okQ[n], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jun 11 2019, after Michel Lagneau *)

PARI

is_A106741(n)=denominator(bernfrac(2*n))%n==0 \\ Charles R Greathouse IV, May 02 2012

PARI

{ for (n=1, 10^6, m = 2*n + 1; for (k=2, n, if ( Mod(k,n)^m != k,  next(2) ); ); print1(n,", "); ); } /* Joerg Arndt, May 04 2012 */

PARI

is_A106741(n)={ my(m=2*n+1); for(k=2, n, Mod(k, n)^m - k & return); 1} /* more than twice faster (in PARI 2.4.2) than with "if(...)" */ \\ M. F. Hasler, May 06 2012

Extensions

Terms a(19)-a(29) from Michel Lagneau, May 02 2012

Terms >= 10230 by Joerg Arndt, May 04 2012