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Search: id:A089162
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| A089162 |
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Prime factors of Mersenne numbers. |
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+0
1
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| 3, 7, 31, 127, 23, 89, 8191, 131071, 524287, 47, 178481, 233, 1103, 2089, 2147483647, 223, 616318177, 13367, 164511353, 431, 9719, 2099863, 2351, 4513, 13264529, 6361, 69431, 20394401, 179951, 3203431780337, 2305843009213693951, 193707721
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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All factors of Mersenne numbers 2^p - 1, where p is prime, are = 1 (mod p). See link for a proof of the statement if q divides M_p = 2^p-1 then q = 2kp + 1 for some integer p.
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LINKS
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R. P. Brent, New factors of Mersenne numbers
Chris Caldwell, MersenneNumbers History.
C. K. Caldwell, The Prime Glossary, Mersenne divisor
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EXAMPLE
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The 16th Mersenne number 2^53-1 has the three prime factors 6361,69431,20394401.
See tail end of second row in the sequence. Each factor is = 1 (mod 53).
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PROGRAM
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(PARI) mersenne(b, n, d) = { c=0; forprime(x=2, n, c++; y = b^x-1; f=factor(y); v=component(f, 1); ln = length(v); if(ln>=d, print1(v[d]", ")); ) }
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CROSSREFS
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Cf. A016047.
Cf. A003260.
Sequence in context: A105765 A061095 A103901 this_sequence A016047 A003260 A138865
Adjacent sequences: A089159 A089160 A089161 this_sequence A089163 A089164 A089165
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KEYWORD
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easy,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Dec 06 2003
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