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A106741 Numbers n such that n divides the denominator of 2n-th Bernoulli number. 5
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified December 9 09:33 EST 2019. Contains 329877 sequences. (Running on oeis4.)