  A composite positive integer $n$ is said to be a {\it weak Carmichael number}
if
  $$
  \sum_{\gcd(k,n)=1\atop 1\le k\le n-1}k^{n-1}\equiv \varphi(n) \pmod{n}.
\leqno(1)
  $$ It is proved that a composite positive integer $n$ is a weak Carmichael
number if and only if $p-1\mid n-1$ for every prime divisor $p$ of $n$. This
together with Korselt's criterion yields the fact that every Carmichael number
is also a weak Carmichael number.
  In this paper we mainly investigate arithmetic properties of weak Carmichael
numbers. Motivated by the investigations of Carmichael numbers in the last
hundred years, here we establish several related results, notions, examples and
computatinoal searches for weak Carmichael numbers and numbers closely related
to weak Carmichael numbers. Furthermore, using the software {\tt Mathematica
8}, we present the table containing all non-prime powers weak Carmichael
numbers less than $2\times 10^6$.
  Motivated by heuristic arguments, our computations and some old conjectures
and results for Carmichael numbers, we propose several conjectures for weak
Carmichael numbers and for some other classes of Carmichael like numbers.
  Finally, we consider weak Carmichael numbers in light of Fermat primality
test. We believe that it can be of interest to involve certain particular
classes of weak Carmichael numbers in some problems concerning Fermat-like
primality tests and the generalized Riemann hypothesis.
