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A287934
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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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0
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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Jozsef Sandor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 5, p. 556.
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LINKS
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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