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A125854
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Primes p with the property that p divides the Wolstenholme number A001008((p+1)/2).
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7
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OFFSET
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1,1
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COMMENTS
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Note that if prime p>3 divides A001008((p+1)/2) then it also divides A001008((p-3)/2).
Note that for a prime p, H([p/2]) == 2*(2^(-p(p-1)) - 1)/p^2 (mod p). Therefore a prime p divides the Wolstenholme number A001008((p+1)/2) if and only if 2^(-p(p-1)) == 1 - p^2 (mod p^3) or, equivalently, 2^(p-1) == 1 + p (mod p^2).
Disjunctive union of the sequences A154998 and A121999 that contain primes congruent respectively to 1,3 and 5,7 modulo 8. (Alekseyev)
Primes p that are base-((p-1)/2) Wieferich primes, that is, primes p such that ((p-1)/2)^(p-1) == 1 (mod p^2). - Jianing Song, Jan 27 2019
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LINKS
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EXAMPLE
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a(1) = 3 because prime 3 divides A001008(2) = 3 and there is no p < 3 that divides A001008((p+1)/2).
a(2) = 29 because 29 divides A001008(15) = 1195757 and there is no prime p (3 < p < 29) that divides A001008((p+1)/2).
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MATHEMATICA
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Select[Prime[Range[1, 5000]],
Divisible[Numerator[HarmonicNumber[(# + 1)/2]], #] &] (* Robert Price, May 10 2019 *)
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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Entry revised and a(5) = 2001907169 provided by Max Alekseyev, Jan 18 2009
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STATUS
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approved
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