OFFSET
1,2
COMMENTS
By Zsigmondy's theorem, the n-th Zsigmondy number for bases a and b is not 1 except in the three cases (1) a = 2, b = 1, n = 1, (2) a = 2, b = 1, n = 6, (3) n = 2 and a + b is a power of 2.
Composite terms are the maximal overpseudoprimes to base 2 (see A141232) for which the multiplicative order of 2 mod a(n) equals n. - Vladimir Shevelev, Aug 26 2008
a(n) = 2^n - 1 if and only if either n = 1 or n is prime. - Vladimir Shevelev, Sep 30 2008
a(n) == 1 (mod n), 2^(a(n)-1) == 1 (mod a(n)), A002326((a(n)-1)/2) = n. - Thomas Ordowski, Oct 25 2017
If n is odd, then the prime factors of a(n) are congruent to {1,7} mod 8, that is, they have 2 has a quadratic residue, and are congruent to 1 mod 2n. If n is divisible by 8, then the prime factors of a(n) are congruent to 1 mod 16. - Jianing Song, Apr 13 2019
Named after the Austrian mathematician Karl Zsigmondy (1867-1925). - Amiram Eldar, Jun 20 2021
LINKS
T. D. Noe, Table of n, a(n) for n=1..1000
Wikipedia, Zsigmondy's theorem.
Karl Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik, Vol. 3 (1892), pp. 265-284.
FORMULA
Denominator of Sum_{d|n} d*moebius(n/d)/(2^d-1). - Vladeta Jovovic, Apr 02 2004
EXAMPLE
a(4) = 5 because 2^4 - 1 = 15 and its divisors being 1, 3, 5, 15, only 1 and 5 are coprime to 2^2 - 1 = 3 and 2^3 - 1 = 7, and 5 is the greater of these.
a(5) = 31 because 2^5 - 1 = 31 is prime.
a(6) = 1 because 2^6 - 1 = 63 and its divisors being 1, 3, 7, 9, 21, 63, only 1 is coprime to all of 3, 7, 15, 31.
MATHEMATICA
Table[Cyclotomic[n, 2]/GCD[n, Cyclotomic[n, 2]], {n, 40}] (* Alonso del Arte, Mar 14 2013 *)
PROG
(PARI) a(n) = my(m = polcyclo(n, 2)); m/gcd(m, n); \\ Michel Marcus, Mar 07 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Jens Voß, Sep 04 2001
EXTENSIONS
More terms from Vladeta Jovovic, Apr 02 2004
Definition corrected by Jerry Metzger, Nov 04 2009
STATUS
approved