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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).
0
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jun 03 2017
STATUS
approved