A298365 Numbers k such that there exists at least one odd pseudoprime of order k.

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

Author

Krzysztof Ziemak, Jan 17 2018

Keywords

• nonn

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.

Examples

			10 is a term since 341 is an odd pseudoprime whose order is 10.
		

Links

• Krzysztof Ziemak, First 172 class [a(n)] of odd pseudoprime numbers
• Krzysztof Ziemak, PARI code for generation sequence a(n)

Crossrefs

Cf. A001567, A086249.

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.