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

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

1,1

Comments

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).

References

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

Links

Table of n, a(n) for n=1..39.

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.

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A235810 A229906 A008367 * A152852 A156572 A157990

Adjacent sequences: A287931 A287932 A287933 * A287935 A287936 A287937

Keyword

nonn

Author

Amiram Eldar, Jun 03 2017

Status

approved