A180942 -id:A180942 - cp's OEIS frontend

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

Vladimir Shevelev, Sep 27 2010

nonn

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.

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]

Cf. A000367, A002445, A180942.

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

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 *)

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

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

A287934 Composite numbers n such that E(n+1)+1 is divisible by n, where E(n) is the n-th Euler number (A122045).

Original entry on oeis.org

289, 341, 561, 1105, 1369, 1387, 1729, 2465, 2821, 4097, 5365, 6179, 6601, 8911, 9105, 9537, 10585, 12673, 14433, 14531, 15457, 15841, 28033, 29341, 33901, 41041, 41905, 42141, 46657, 48705, 52633, 52741, 62745, 63253, 63973, 75361, 80185, 82621, 99937

Offset: 1

Amiram Eldar, Jun 03 2017

nonn

Kummer proved in 1851 that E(2k + p - 1) == E(2k) (mod p) for k > 0 and all odd primes p. This sequence consists of composite numbers for which the congruence, with k=1, also holds. In terms of A000364, the sequence consists of composite odd numbers n that divide A000364((n + 1)/2) + (-1)^((n + 1)/2).

Jozsef Sandor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 5, p. 556.

Leonard Carlitz, Congruences for generalized Bell and Stirling numbers, Duke Mathematical Journal, Vol. 22, No. 2 (1955), pp. 193-205.
Ernst Eduard Kummer, Über eine allgemeine Eigenschaft der rationalen Entwickelungscoefficienten einer bestimmten Gattung analytischer Functionen, Journal für die reine und angewandte Mathematik, Vol. 41 (1851), pp. 368-372.
Samuel S. Wagstaff, Jr., Prime divisors of the Bernoulli and Euler numbers, Number Theory for the Millenium III (Urbana, IL, 2000), AK Peters, Natick, MA, 2002, pp. 357-374.

Cf. A000364, A035163, A081730, A122045, A180942.

a={}; For[n = 1, n < 100000, n++; If[!PrimeQ[n] && Divisible[EulerE[n + 1] + 1, n], a=AppendTo[a,n]]];a
Select[Range[100000],CompositeQ[#]&&Divisible[EulerE[#+1]+1,#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 03 2019 *)

e(n) = 2^n*2^(n+1)*(subst(bernpol(n+1, x), x, 3/4) - subst(bernpol(n+1, x), x, 1/4))/(n+1);
isok(n) = (((e(n+1)+1) % n) == 0);
lista(nn) = forcomposite(n=1, nn, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Jun 10 2017

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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).

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A287934 Composite numbers n such that E(n+1)+1 is divisible by n, where E(n) is the n-th Euler number (A122045).

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cp's OEIS Frontend

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

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A287934 Composite numbers n such that E(n+1)+1 is divisible by n, where E(n) is the n-th Euler number (A122045).

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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).