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

1,1

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

Table of n, a(n) for n=1..78.

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

Cf. A001567, A086249.

Sequence in context: A095038 A085186 A345289 * A335237 A214864 A062844

Adjacent sequences: A298362 A298363 A298364 * A298366 A298367 A298368

Keyword

nonn

Author

Krzysztof Ziemak, Jan 17 2018

Status

approved