A392903 Irregular triangle read by rows: row n lists the distinct prime factors of the generalized Fermat number F_n(7) = 7^(2^n) + 1.

2, 2, 5, 2, 1201, 2, 17, 169553, 2, 353, 47072139617, 2, 7699649, 134818753, 531968664833, 2, 35969, 1110623386241, 15266848196793556098085000332888634369, 2, 257, 769, 197231873, 6856531741041792239054980342217258517995521, 2783415704056554985941269027566547436008362462334209

Offset

0,1

Comments

No F_n(7) number is prime.

F_n(7)/2 is currently known to be prime only for n = 2.

Links

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

Wilfrid Keller, Prime factors of generalized Fermat numbers F'_m(7) = F_m(7)/2 and complete factoring status.

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

Wikipedia, Generalized Fermat numbers.

Example

Triangle begins:
     | F_n(7) =   |
   n | A078304(n) | Distinct prime factors of F_n(7)
  ----------------------------------------------------------------------------------
   0 |    7^1 + 1 | 2;
   1 |    7^2 + 1 | 2, 5;
   2 |    7^4 + 1 | 2, 1201;
   3 |    7^8 + 1 | 2, 17, 169553;
   4 |   7^16 + 1 | 2, 353, 47072139617;
   5 |   7^32 + 1 | 2, 7699649, 134818753, 531968664833;
   6 |   7^64 + 1 | 2, 35969, 1110623386241, 15266848196793556098085000332888634369;
  ...

Mathematica

A392903row[n_] := FactorInteger[7^2^n + 1][[All, 1]];
Array[A392903row, 7, 0]

Crossrefs

Cf. A078304, A253242.

Cf. A050922 (b=2), A392900 (b=3), A392901 (b=5), A392902 (b=6), A393152 (b=8), A391444 (b=10), A392904 (b=11), A392905 (b=12).

Sequence in context: A284464 A038041 A392900 * A197591 A097891 A097611

Adjacent sequences: A392900 A392901 A392902 * A392904 A392905 A392906

Keyword

nonn,tabf,hard

Author

Paolo Xausa, Jan 27 2026

Status

approved