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

33, 169, 481, 561, 793, 805, 949, 1105, 1261, 1417, 1645, 1729, 2041, 2353, 2465, 2509, 2821, 2977, 3133, 3421, 3445, 3601, 4069, 4123, 4381, 4537, 4849, 5161, 5317, 5473, 5629, 5941, 6061, 6205, 6601, 7033, 7093, 7189, 7501, 7813, 7885, 7969, 8113

Offset: 1

Author

Vladimir Shevelev, Sep 27 2010

Keywords

• nonn

Comments

These are pseudoprimes in the sense that the congruence of the definition is valid if any odd prime is substituted for m.

Entries of the form m = 4*k+3 are apparently rare: 4123, 8911, ...

Computed to 50 terms by D. S. McNeil, Sep 05 2010.

Links

• V. Shevelev, B-pseudoprimes, seqfan list, Sep 04 2010
• Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS, 12 (2012), #A1. [_N. J. A. Sloane_, Feb 07 2013]

Crossrefs

Cf. A000367, A002445, A180942.

Programs

Maple

A000367 := proc(n) numer(bernoulli(2*n)) ; end proc:
A002445 := proc(n) denom(bernoulli(2*n)) ; end proc:
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:
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

Mathematica

nb[n_] := Numerator[BernoulliB[2n]];
db[n_] := Denominator[BernoulliB[2n]];
okQ[m_] := CompositeQ[m] && Mod[12*Abs[nb[(m+1)/2]], m] == Mod[(-1)^((m-1)/2)*db[(m+1)/2], m];
Select[Range[33, 9999, 2], okQ] (* Jean-François Alcover, Feb 28 2024 *)

Extensions

Comments rephrased and program added by R. J. Mathar, Oct 24 2010

Typo in data fixed by Jean-François Alcover, Feb 28 2024