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A180943 Odd composite numbers m for which 12*|A000367((m+1)/2)|==(-1)^{(m-1)/ 2}* A002445((m+1)/2) (mod m).

%I A180943 #20 Dec 23 2024 14:53:42
%S A180943 33,169,481,561,793,805,949,1105,1261,1417,1645,1729,2041,2353,2465,
%T A180943 2509,2821,2977,3133,3421,3445,3601,4069,4123,4381,4537,4849,5161,
%U A180943 5317,5473,5629,5941,6061,6205,6601,7033,7093,7189,7501,7813,7885,7969,8113
%N A180943 Odd composite numbers m for which 12*|A000367((m+1)/2)|==(-1)^{(m-1)/ 2}* A002445((m+1)/2) (mod m).
%C A180943 These are pseudoprimes in the sense that the congruence of the definition is valid if any odd prime is substituted for m.
%C A180943 Entries of the form m = 4*k+3 are apparently rare: 4123, 8911, ...
%C A180943 Computed to 50 terms by _D. S. McNeil_, Sep 05 2010.
%H A180943 V. Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2010-September/005904.html">B-pseudoprimes</a>, seqfan list, Sep 04 2010
%H A180943 Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/m1/m1.Abstract.html">The number of permutations with prescribed up-down structure as a function of two variables</a>, INTEGERS, 12 (2012), #A1. [_N. J. A. Sloane_, Feb 07 2013]
%p A180943 A000367 := proc(n) numer(bernoulli(2*n)) ; end proc:
%p A180943 A002445 := proc(n) denom(bernoulli(2*n)) ; end proc:
%p A180943 isA180943 := proc(m) if type(m,'odd') and not isprime(m) then 12*abs(A000367((m+1)/2)) mod m = (-1)^((m-1)/2)*A002445((m+1)/2) mod m ; else false; end if; end proc:
%p A180943 A180943 := proc(n) option remember; if n = 1 then 33; else for a from procname(n-1)+2 by 2 do if isA180943(a) then return a; end if; end do: end if; end proc: # _R. J. Mathar_, Oct 24 2010
%t A180943 nb[n_] := Numerator[BernoulliB[2n]];
%t A180943 db[n_] := Denominator[BernoulliB[2n]];
%t A180943 okQ[m_] := CompositeQ[m] && Mod[12*Abs[nb[(m+1)/2]], m] == Mod[(-1)^((m-1)/2)*db[(m+1)/2], m];
%t A180943 Select[Range[33, 9999, 2], okQ] (* _Jean-François Alcover_, Feb 28 2024 *)
%Y A180943 Cf. A000367, A002445, A180942.
%K A180943 nonn
%O A180943 1,1
%A A180943 _Vladimir Shevelev_, Sep 27 2010
%E A180943 Comments rephrased and program added by _R. J. Mathar_, Oct 24 2010
%E A180943 Typo in data fixed by _Jean-François Alcover_, Feb 28 2024