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A089162
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Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.
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3
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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
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OFFSET
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1,1
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COMMENTS
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All factors of Mersenne numbers 2^p - 1, where p is prime, are = 1 (mod p). See the first Caldwell link for a proof of the statement if q divides M_p = 2^p-1 then q = 2kp + 1 for some integer k. - Comment corrected by Jonathan Sondow, Dec 29 2016
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LINKS
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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).
Triangle begins:
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;
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PROG
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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. A122094 (sorted version of this list).
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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