Editing Fermat pseudoprime (section)

==Weak pseudoprimes==
A composite number ''n'' which satisfies <math>b^n \equiv b \pmod n</math> is called a '''weak pseudoprime to base ''b'''''.  For any given base ''b'', all Fermat pseudoprimes are weak pseudoprimes, and all weak pseudoprimes coprime to ''b'' are Fermat pseudoprimes.  However, this definition also permits some pseudoprimes which are ''not'' coprime to ''b''.<ref>{{cite web |url=http://www.numericana.com/answer/pseudo.htm#weak |title=Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality |first=Gérard P. |last=Michon |website=Numericana |date=19 November 2003 |access-date=21 April 2018}}</ref>  For example, the smallest even weak pseudoprime to base 2 is 161038 (see {{oeis|id=A006935}}).

The least weak pseudoprime to bases ''b'' = 1, 2, ... are:
:4, 341, 6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, 9, 6, 4, 4, 6, 6, 4, 4, 6, 9, 4, 4, 38, 6, 4, 4, 6, 6, 4, 4, 6, 46, 4, 4, 10, ... {{OEIS2C|id=A000790}}

Carmichael numbers are weak pseudoprimes to all bases, thus all terms in this list are less than or equal to the smallest Carmichael number, 561. Except for 561 = 3⋅11⋅17, only [[semiprime]]s can occur in the above sequence. Not all semiprimes less than 561 occur; a semiprime ''pq'' (''p'' ≤ ''q'') less than 561 occurs in the above sequences [[if and only if]] ''p'' − 1 divides ''q'' − 1 (see {{oeis|id=A108574}}).  The least Fermat pseudoprime to base ''b'' (also not necessary exceeding ''b'') ({{oeis|A090086}}) is ''usually'' semiprime, but not always; the first counterexample is {{OEIS link|A090086}}(648) = 385 = 5 × 7 × 11.

If we require ''n'' > ''b'', the least weak pseudoprimes (for ''b'' = 1, 2, ...) are:
:4, 341, 6, 6, 10, 10, 14, 9, 12, 15, 15, 22, 21, 15, 21, 20, 34, 25, 38, 21, 28, 33, 33, 25, 28, 27, 39, 36, 35, 49, 49, 33, 44, 35, 45, 42, 45, 39, 57, 52, 82, 66, 77, 45, 55, 69, 65, 49, 56, 51, ... {{OEIS2C|id=A239293}}