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A064078
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Zsigmondy numbers for a = 2, b = 1: Zs(n, 2, 1) is the greatest divisor of 2^n - 1^n (A000225) that is relatively prime to 2^m - 1^m for all positive integers m < n.
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17
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1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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 a(n) are the maximal overpseudoprimes to base 2 (see A141232) for which the multiplicative order of 2 mod a(n) equals n. - Vladimir Shevelev (shevelev(AT)bgu.ac.il), Aug 26 2008
a(n)=2^n-1 iff either n=1 or n is prime [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Sep 30 2008]
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REFERENCES
| K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte fuer Mathematik und Physik 3 (1882), 265 - 284
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. f. Math. III. (1892) 265-284.
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FORMULA
| Denominator of Sum_{d|n} d*moebius(n/d)/(2^d-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 02 2004
Phi(n,2)/gcd(n,Phi(n,2)), where Phi(n,2) is the n-th cyclotomic polynomial evaluated at 2. [From T. D. Noe (noe(AT)sspectra.com), Apr 13 2010]
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CROSSREFS
| Cf. A000225, A064079 - A064083.
Cf. A019320, A063982.
Sequence in context: A046561 A097406 A112927 * A186522 A048857 A005420
Adjacent sequences: A064075 A064076 A064077 * A064079 A064080 A064081
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KEYWORD
| nonn
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AUTHOR
| Jens Voss (jens.voss(AT)poet.de), Sep 04 2001
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 02 2004
Definition corrected by Jerry Metzger, Nov 04 2009
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