[seqfan] Reduced Bernoulli numbers
Tomasz Ordowski
tomaszordowski at gmail.com
Mon Jul 13 10:23:57 CEST 2020
Dear readers!
Let F(n) = Sum_{p prime, p-1|n} 1/p = N(n) / D(n), for n > 0.
By von Staudt-Clausen theorem, B(2n) + F(2n) is an integer,
namely A000146(n) : https://oeis.org/A000146
Conjecture:
p N(p-1) == D(p-1) (mod p^2) if and only if p is prime.
Question: Is this equivalent to the Agoh-Giuga conjecture?
The weak pseudoprimes (mod p) are Carmichael numbers.
I also noticed that
a composite n is a Giuga number
if and only if n N(phi(n)) == D(phi(n)) (mod n^2).
Euler's phi function can be replaced by the Carmichael lambda function.
Best regards,
Thomas Ordowski
____________________________
F(n) = N(n) / D(n) = A027759(n) / A027760(n) :
https://oeis.org/A027759 / https://oeis.org/A027760
https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture
https://en.wikipedia.org/wiki/Von_Staudt%E2%80%93Clausen_theorem
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