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

Author

Amiram Eldar, Jun 03 2017

Keywords

• nonn

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

• 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.

Crossrefs

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

Programs

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 *)

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