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

1,1

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

Table of n, a(n) for n=1..43.

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]

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 *)

Crossrefs

Cf. A000367, A002445, A180942.

Sequence in context: A183776 A256021 A172077 * A113752 A155883 A071914

Adjacent sequences: A180940 A180941 A180942 * A180944 A180945 A180946

Keyword

nonn,changed

Author

Vladimir Shevelev, Sep 27 2010

Extensions

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

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

Status

approved