A089162 - OEIS

%I #38 May 03 2024 20:19:46

%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 13264529,6361,69431,20394401,179951,3203431780337,2305843009213693951,193707721,761838257287

%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 that 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 K. Caldwell, <a href="https://t5k.org/mersenne/index.html">Mersenne Primes: History, Theorems and Lists</a>.

%H Chris 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;

%t row[n_]:=First/@FactorInteger[2^Prime[n]-1]; Array[row,19]//Flatten (* _Stefano Spezia_, May 03 2024 *)

%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