A392900 Irregular triangle read by rows: row n lists the distinct prime factors of the generalized Fermat number F_n(3) = 3^(2^n) + 1.

2, 2, 5, 2, 41, 2, 17, 193, 2, 21523361, 2, 926510094425921, 2, 1716841910146256242328924544641, 2, 257, 275201, 138424618868737, 3913786281514524929, 153849834853910661121, 2, 12289, 8972801, 891206124520373602817, 707275264749309881405141965802671548079179711820351316861777644606207216944972589404100097

Offset

0,1

Comments

No F_n(3) number is prime.

F_n(3)/2 is currently known to be prime only for n in {0, 1, 2, 4, 5, 6}.

Links

Table of n, a(n) for n=0..24.

Wilfrid Keller, Prime factors of generalized Fermat numbers F'_m(3) = F_m(3)/2 and complete factoring status.

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

Wikipedia, Generalized Fermat numbers.

Example

Triangle begins:
     | F_n(3) =   |
   n | A059919(n) | Distinct prime factors of F_n(3)
  -----------------------------------------------------
   0 |    3^1 + 1 | 2;
   1 |    3^2 + 1 | 2, 5;
   2 |    3^4 + 1 | 2, 41;
   3 |    3^8 + 1 | 2, 17, 193;
   4 |   3^16 + 1 | 2, 21523361;
   5 |   3^32 + 1 | 2, 926510094425921;
   6 |   3^64 + 1 | 2, 1716841910146256242328924544641;
  ...

Mathematica

A392900row[n_] := FactorInteger[3^2^n + 1][[All, 1]];
Array[A392900row, 9, 0]

Crossrefs

Cf. A059919, A253242.

Cf. A050922 (b=2), A392901 (b=5), A392902 (b=6), A392903 (b=7), A393152 (b=8), A391444 (b=10), A392904 (b=11), A392905 (b=12).

Sequence in context: A226135 A284464 A038041 * A392903 A197591 A097891

Adjacent sequences: A392897 A392898 A392899 * A392901 A392902 A392903

Keyword

nonn,tabf,hard

Author

Paolo Xausa, Jan 26 2026

Status

approved