OFFSET
1,1
COMMENTS
A composite divisor d of M(m) := 2^m - 1 is called primitive if M(k) != 0 for any k < m.
A primitive composite divisor d of M(m) is said to have rank m, and we write rank(d)=m.
Let M(m)=2^m-1, and define D to be the set of all numbers d such that d|M(m), d==1 (mod m), and rank(d)=m. Then each element d from D is an odd pseudoprime, because if m|d-1, then M(m)|M(d-1) and thus d|M(d-1). The set D contains all composite and primitive divisors d|M(m) that have rank(d)=m and each odd pseudoprime d with rank(d)=m generates only one class [a(n)] with all pseudoprimes d, where a(n)=m, if a(n) is defined as below. See attached file with examples of pseudoprimes.
LINKS
Krzysztof Ziemak, First 172 class [a(n)] of odd pseudoprime numbers
Krzysztof Ziemak, PARI code for generation sequence a(n)
FORMULA
a(n) = min{k: k>a(n-1) and M(k) has a composite divisor d and rank(d)=k and d==1 (mod k)} for n = 1,2,3,... where M(k):=2^k-1.
EXAMPLE
10 is a term since 341 is an odd pseudoprime whose order is 10.
CROSSREFS
KEYWORD
nonn
AUTHOR
Krzysztof Ziemak, Jan 17 2018
STATUS
approved