A392901 Irregular triangle read by rows: row n lists the distinct prime factors of the generalized Fermat number F_n(5) = 5^(2^n) + 1.

2, 3, 2, 13, 2, 313, 2, 17, 11489, 2, 2593, 29423041, 2, 641, 75068993, 241931001601, 2, 769, 3666499598977, 96132956782643741951225664001, 2, 257, 23653200983830003298459393, 24171717725330873572798545219226642215966994254472458802413313, 2, 1655809, 101199664791578113, 4563566430220614493697, 12025702000065183805751513732616276516181800961, 4695555596393879529570509800156232075658173303955183176829795300834096134412809302652417

Offset

0,1

Comments

No F_n(5) number is prime.

F_n(5)/2 is currently known to be prime only for n <= 2.

Links

Table of n, a(n) for n=0..29.

Wilfrid Keller, Prime factors of generalized Fermat numbers F'_m(5) = F_m(5)/2 and complete factoring status.

Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Wikipedia, Generalized Fermat numbers.

Example

Triangle begins:
     | F_n(5) =   |
   n | A199591(n) | Distinct prime factors of F_n(5)
  -----------------------------------------------------------------------
   0 |    5^1 + 1 | 2, 3;
   1 |    5^2 + 1 | 2, 13;
   2 |    5^4 + 1 | 2, 313;
   3 |    5^8 + 1 | 2, 17, 11489;
   4 |   5^16 + 1 | 2, 2593, 29423041;
   5 |   5^32 + 1 | 2, 641, 75068993, 241931001601;
   6 |   5^64 + 1 | 2, 769, 3666499598977, 96132956782643741951225664001;
  ...

Mathematica

A392901row[n_] := FactorInteger[5^2^n + 1][[All, 1]];
Array[A392901row, 7, 0]

Crossrefs

Cf. A199591, A253242.

Cf. A050922 (b=2), A392900 (b=3), A392902 (b=6), A392903 (b=7), A393152 (b=8), A391444 (b=10), A392904 (b=11), A392905 (b=12).

Sequence in context: A012960 A013117 A120864 * A173463 A023641 A083775

Adjacent sequences: A392898 A392899 A392900 * A392902 A392903 A392904

Keyword

nonn,tabf,hard

Author

Paolo Xausa, Jan 26 2026

Status

approved