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A298365
Numbers k such that there exists at least one odd pseudoprime of order k.
1
10, 11, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97
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.
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
Sequence in context: A095038 A085186 A345289 * A373634 A373633 A373665
KEYWORD
nonn
AUTHOR
Krzysztof Ziemak, Jan 17 2018
STATUS
approved