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

0,1

Comments

Rows have irregular lengths.

The length of row n is A054992(n).

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.

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A321882 A281158 A100742 * A201769 A077276 A073684

Adjacent sequences: A001266 A001267 A001268 * A001270 A001271 A001272

Keyword

nonn,tabf

Author

N. J. A. Sloane

Status

approved