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A125854 Primes p with the property that p divides the Wolstenholme number A001008((p+1)/2). 7
3, 29, 37, 3373, 2001907169 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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)
a(6) > 5.5*10^12. - Giovanni Resta, Apr 13 2017
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
LINKS
EXAMPLE
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).
MATHEMATICA
Select[Prime[Range[1, 5000]],
Divisible[Numerator[HarmonicNumber[(# + 1)/2]], #] &] (* Robert Price, May 10 2019 *)
CROSSREFS
Sequence in context: A086174 A338518 A178642 * A167278 A341210 A106979
KEYWORD
hard,more,nonn
AUTHOR
Alexander Adamchuk, Dec 11 2006
EXTENSIONS
Entry revised and a(5) = 2001907169 provided by Max Alekseyev, Jan 18 2009
Edited by Max Alekseyev, Oct 13 2009
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)