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Search: id:A064078
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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
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OFFSET
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1,2
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COMMENT
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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
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K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte fuer Mathematik und Physik 3 (1882), 265 - 284
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. f. Math. III. 265-284. Pub(1892)
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FORMULA
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Denominator of Sum_{d|n} d*moebius(n/d)/(2^d-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 02 2004
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CROSSREFS
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Cf. A000225, A064079, A064080, A064081, A064082, A064083.
Cf. A019320, A063982.
Sequence in context: A046561 A097406 A112927 this_sequence A048857 A005420 A161818
Adjacent sequences: A064075 A064076 A064077 this_sequence A064079 A064080 A064081
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KEYWORD
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nonn
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AUTHOR
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Jens Voss (jens.voss(AT)poet.de), Sep 04 2001
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 02 2004
Definition corrected by Jerry Metzger, Nov 04 2009
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