A298365 - OEIS

%I #130 Feb 15 2018 07:48:30

%S 10,11,14,15,16,18,20,21,22,23,24,25,26,28,29,30,32,33,34,35,36,37,38,

%T 39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,57,58,59,60,62,63,

%U 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

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

%C A composite divisor d of M(m) := 2^m - 1 is called primitive if M(k) != 0 for any k < m.

%C A primitive composite divisor d of M(m) is said to have rank m, and we write rank(d)=m.

%C 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.

%H Krzysztof Ziemak, <a href="/A298365/a298365_1.txt">First 172 class [a(n)] of odd pseudoprime numbers</a>

%H Krzysztof Ziemak, <a href="/A298365/a298365_3.txt">PARI code for generation sequence a(n)</a>

%F 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.

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

%Y Cf. A001567, A086249.

%K nonn

%O 1,1

%A _Krzysztof Ziemak_, Jan 17 2018