A001269 Table T(n,k) in which n-th row lists prime factors of 2^n + 1 (n >= 0), with repetition.

2, 3, 5, 3, 3, 17, 3, 11, 5, 13, 3, 43, 257, 3, 3, 3, 19, 5, 5, 41, 3, 683, 17, 241, 3, 2731, 5, 29, 113, 3, 3, 11, 331, 65537, 3, 43691, 5, 13, 37, 109, 3, 174763, 17, 61681, 3, 3, 43, 5419, 5, 397, 2113, 3, 2796203, 97, 257, 673, 3, 11, 251, 4051

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf

Comments

Rows have irregular lengths.

The length of row n is A054992(n).

Examples

			Triangle begins:
  2;
  3;
  5;
  3,3,17;
  3,11;
  5,13;
  3,43;
  257;
  ...
		

References

• J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Links

• Max Alekseyev, Rows n = 0..1122, flattened (rows 0..500 from T. D. Noe)
• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
• S. S. Wagstaff, Jr., The Cunningham Project
• Chai Wah Wu, Tables from the Cunningham Project in machine-readable JSON format.

Crossrefs

Cf. A060444 (factors w/o repetition), A054992 (row lengths).

Programs

Mathematica

repeat[{p_, e_}] := Table[p, {e}]; row[n_] := repeat /@ FactorInteger[2^n + 1] // Flatten; Table[row[n], {n, 0, 25}] // Flatten (* Jean-François Alcover, Jul 13 2012 *)

PARI

apply( A001269_row(n)=concat(apply(f->vector(f[2],i,f[1]), Col(factor(2^n+1))~)), [0..19]) \\ M. F. Hasler, Nov 19 2018