A Carmichael number $n$ has the property that
$$a^{n-1}\equiv 1\ (\ mod\ n\ )$$
for every $a$ with $(a,n)=1$.
I wonder, which numbers have the converse property :
For which composite numbers $n$, every number $a$ with $1<a<n-1$ is a witness ?
That means, for which composite numbers $n$ does $a^{n-1}\neq 1\ (\ mod\ n\ )$ hold for every $a=2,3,...,n-3,n-2$ ?
The numbers upto $200$ with this property are :
? for(n=2,200,if(isprime(n)==0,gef=1;for(a=2,n-2,if(Mod(a^(n-1),n)==1,gef=0));if
(gef==1,print1(n," "))))
4 6 8 9 10 12 14 16 18 20 22 24 26 27 30 32 34 36 38 40 42
44 46 48 50 54 56 58 60 62 64 68 72 74 78 80 81 82 84 86 88
90 92 94 96 98 100 102 104 106 108 110 114 116 118 120 122 126
128 132 134 136 138 140 142 144 146 150 152 156 158 160 162 164
166 168 170 174 178 180 182 184 188 192 194 198 200
?
The first few odd composite numbers with this property :
? forstep(n=3,2000,2,if(isprime(n)==0,gef=1;for(a=2,n-2,if(Mod(a^(n-1),n)==1,gef
=0));if(gef==1,print1(n," "))))
9 27 81 243 729
So, from the odd composite numbers, only the powers of $3$ seem to have this property.
The first few even composite numbers, NOT having this property :
? forstep(n=2,100,2,if(isprime(n)==0,gef=1;for(a=2,n-2,if(Mod(a^(n-1),n)==1,gef=
0));if(gef==0,print1(n," "))))
28 52 66 70 76