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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-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

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