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

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, 761838257287

Offset: 1

Author

Cino Hilliard, Dec 06 2003

Keywords

• nonn
• tabf

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 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

Examples

			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;
		

Links

• Max Alekseyev, Rows n = 1..197, flattened (rows 1..167 from Jens Kruse Andersen)
• R. P. Brent, New factors of Mersenne numbers.
• Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists.
• Chris K. Caldwell, The Prime Glossary, Mersenne divisor.
• Sam Wagstaff, The Cunningham Project.

Crossrefs

Cf. A001348, A003260, A016047, A088863.

Cf. A122094 (sorted version of this list).

Programs

Mathematica

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

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]",")); ) }

Extensions

Definition corrected by Max Alekseyev, Jul 25 2023