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A089162 Prime factors of Mersenne numbers. 3
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; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Jens Kruse Andersen, Table of n, a(n) for n = 1..591

R. P. Brent, New factors of Mersenne numbers

Chris Caldwell, MersenneNumbers History.

C. K. Caldwell, The Prime Glossary, Mersenne divisor

Sam Wagstaff, The Cunningham Project

EXAMPLE

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;

PROG

(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]", ")); ) }

CROSSREFS

Cf. A001348, A003260, A016047.

Cf. A122094 (sorted version of this list).

Sequence in context: A105765 A061095 A103901 * A016047 A003260 A152058

Adjacent sequences:  A089159 A089160 A089161 * A089163 A089164 A089165

KEYWORD

nonn,tabf

AUTHOR

Cino Hilliard, Dec 06 2003

STATUS

approved

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Last modified August 10 22:22 EDT 2020. Contains 336403 sequences. (Running on oeis4.)