OFFSET
1,1
COMMENTS
A064078(a(n)) is a composite number. The sequence has a positive density since it contains, in particular, numbers of the form 8n+20 for n >= 1 (C. Pomerance, private correspondence). Since, e.g., 38 is not in the sequence, there is not an overpseudoprime m such that ord_m(2)=38.
Phi_{a(n)}(2), the a(n)-th cyclotomic polynomial of x evaluated at x=2 has at least 2 distinct prime factors that are not prime factors of the Phi_k(2) for any positive integer k < a(n). For example, Phi_11(2) = 2^11 - 1 = 2047 = 23 * 89 and Phi_25(2) = 2^20 + 2^15 + 2^10 + 2^5 + 1 = 1082401 = 601 * 1801. Note that p = a(n) is prime if and only if Phi_p(2) = 2^p - 1 is composite. - David Terr, Sep 09 2018
It is easy to prove the statement above. We use the fact that Phi_j(n) and Phi_k(n) are coprime whenever j and k are coprime as well as the fact that an overpseudoprime has at least 2 distinct prime factors. - David Terr, Oct 10 2018
A number k is included iff either 2^k-1 has more than one primitive prime factor (cf. A086251, A161508) or the only primitive prime factor of 2^k-1 is a Wieferich prime (no examples known). - Jeppe Stig Nielsen, Sep 01 2020
LINKS
Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1000
PROG
(PARI) isok(k) = my(m=polcyclo(k, 2)); m/=gcd(m, k); m!=1&&!isprime(m) \\ Jeppe Stig Nielsen, Sep 01 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Aug 25 2008
EXTENSIONS
Name edited by Michel Marcus, Oct 06 2018
More terms from Michel Marcus, Oct 11 2018
Data for terms >= 100 corrected by Jeppe Stig Nielsen, Sep 01 2020
STATUS
approved