[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

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

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