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

1,2

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

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 *)

Prog

(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

Crossrefs

Cf. A002445, A014117.

Sequence in context: A052843 A120707 A047111 * A068991 A187491 A178852

Adjacent sequences: A106738 A106739 A106740 * A106742 A106743 A106744

Keyword

nonn

Author

Benoit Cloitre, May 15 2005

Extensions

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

Terms >= 10230 by Joerg Arndt, May 04 2012

Status

approved