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A064384
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Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!.
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8
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OFFSET
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1,1
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COMMENTS
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If p is in the sequence then p divides 0!-1!+2!-3!+...+(-1)^N N! for all sufficiently large N. Naive heuristics suggest that the sequence should be infinite but very sparse.
A prime p is in the sequence if and only if p|A(p-1), where A(0) = 1 and A(n) = n*A(n-1)+1 = A000522(n). - Jonathan Sondow, Dec 22 2006
Michael Mossinghoff has calculated that 2, 5, 13, 37, 463 are the only terms up to 150 million. - Jonathan Sondow, Jun 12 2007
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REFERENCES
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R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B43.
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LINKS
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EXAMPLE
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5 is in the sequence because 5 is prime and it divides 0!-1!+2!-3!+4!=20.
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MATHEMATICA
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Select[Select[Range[500], PrimeQ], (Mod[Sum[(-1)^(p - 1)*p!, {p, 2, # - 1}], #] == 0) &] (* Julien Kluge, Feb 13 2016 *)
a[0] = 1; a[n_] := a[n] = n*a[n - 1] + 1; Select[Select[Range[500], PrimeQ], (Mod[a[# - 1], #] == 0) &] (* Julien Kluge, Feb 13 2016 with the sequence approach suggested by Jonathan Sondow *)
Select[Prime[Range[500]], Divisible[AlternatingFactorial[#]-1, #]&] (* Harvey P. Dale, Jan 08 2021 *)
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PROG
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(PARI) A=1; for(n=1, 1000, if(isprime(n), if(Mod(A, n)==0, print(n))); A=n*A+1) \\ Jonathan Sondow, Dec 22 2006
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CROSSREFS
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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Kevin Buzzard (buzzard(AT)ic.ac.uk), Sep 28 2001
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EXTENSIONS
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STATUS
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approved
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