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%I #31 Jul 26 2023 01:40:00
%S 3,7,31,127,23,89,8191,131071,524287,47,178481,233,1103,2089,
%T 2147483647,223,616318177,13367,164511353,431,9719,2099863,2351,4513,
%U 13264529,6361,69431,20394401,179951,3203431780337,2305843009213693951,193707721
%N Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.
%C 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
%H Max Alekseyev, <a href="/A089162/b089162.txt">Rows n = 1..197, flattened</a> (rows 1..167 from Jens Kruse Andersen)
%H R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub067.html">New factors of Mersenne numbers</a>.
%H Chris Caldwell, <a href="https://t5k.org/mersenne/index.html">Mersenne Primes: History, Theorems and Lists</a>.
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/MersenneDivisor.html">Mersenne divisor</a>.
%H Sam Wagstaff, <a href="http://homes.cerias.purdue.edu/~ssw/cun/">The Cunningham Project</a>.
%e The 16th Mersenne number 2^53-1 has the three prime factors 6361, 69431, 20394401.
%e See tail end of second row in the sequence. Each factor is = 1 (mod 53).
%e Triangle begins:
%e 3;
%e 7;
%e 31;
%e 127;
%e 23, 89;
%e 8191;
%e 131071;
%e 524287;
%e 47, 178481;
%e 233, 1103, 2089;
%e 2147483647;
%e 223, 616318177;
%e 13367, 164511353;
%e 431, 9719, 2099863;
%e 2351, 4513, 13264529;
%e 6361, 69431, 20394401;
%o (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]",")); ) }
%Y Cf. A001348, A003260, A016047.
%Y Cf. A122094 (sorted version of this list).
%K nonn,tabf
%O 1,1
%A _Cino Hilliard_, Dec 06 2003
%E Definition corrected by _Max Alekseyev_, Jul 25 2023
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