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A089162 Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.

%I A089162 #40 Nov 15 2024 21:04:13
%S A089162 3,7,31,127,23,89,8191,131071,524287,47,178481,233,1103,2089,
%T A089162 2147483647,223,616318177,13367,164511353,431,9719,2099863,2351,4513,
%U A089162 13264529,6361,69431,20394401,179951,3203431780337,2305843009213693951,193707721,761838257287
%N A089162 Triangle read by rows formed by the prime factors of Mersenne number 2^prime(n) - 1, n >= 1.
%C A089162 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 A089162 Max Alekseyev, <a href="/A089162/b089162.txt">Rows n = 1..197, flattened</a> (rows 1..167 from Jens Kruse Andersen)
%H A089162 R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/pub/pub067.html">New factors of Mersenne numbers</a>.
%H A089162 Chris K. Caldwell, <a href="https://t5k.org/mersenne/index.html">Mersenne Primes: History, Theorems and Lists</a>.
%H A089162 Chris K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/MersenneDivisor.html">Mersenne divisor</a>.
%H A089162 Sam Wagstaff, <a href="http://homes.cerias.purdue.edu/~ssw/cun/">The Cunningham Project</a>.
%e A089162 The 16th Mersenne number 2^53-1 has the three prime factors 6361, 69431, 20394401.
%e A089162 See tail end of second row in the sequence. Each factor is == 1 (mod 53).
%e A089162 Triangle begins:
%e A089162   3;
%e A089162   7;
%e A089162   31;
%e A089162   127;
%e A089162   23, 89;
%e A089162   8191;
%e A089162   131071;
%e A089162   524287;
%e A089162   47, 178481;
%e A089162   233, 1103, 2089;
%e A089162   2147483647;
%e A089162   223, 616318177;
%e A089162   13367, 164511353;
%e A089162   431, 9719, 2099863;
%e A089162   2351, 4513, 13264529;
%e A089162   6361, 69431, 20394401;
%t A089162 row[n_]:=First/@FactorInteger[2^Prime[n]-1]; Array[row,19]//Flatten (* _Stefano Spezia_, May 03 2024 *)
%o A089162 (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 A089162 Cf. A001348, A003260, A016047, A088863.
%Y A089162 Cf. A122094 (sorted version of this list).
%K A089162 nonn,tabf
%O A089162 1,1
%A A089162 _Cino Hilliard_, Dec 06 2003
%E A089162 Definition corrected by _Max Alekseyev_, Jul 25 2023