A274903 Largest prime factor of 4^n + 1.

2, 5, 17, 13, 257, 41, 241, 113, 65537, 109, 61681, 2113, 673, 1613, 15790321, 1321, 6700417, 26317, 38737, 525313, 4278255361, 14449, 2931542417, 30269, 22253377, 268501, 308761441, 279073, 54410972897, 536903681, 4562284561, 384773, 67280421310721

Offset: 0

Author

Vincenzo Librandi, Jul 11 2016

Keywords

• nonn

Examples

			4^3 + 1 = 65 = 5*13, so a(3) = 13.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..583
• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Crossrefs

Cf. largest prime factor of k^n+1: A002587 (k=2), A074476 (k=3), this sequence (k=4), A074478 (k=5), A274904 (k=6), A227575 (k=7), A274905 (k=8), A002592 (k=9), A003021 (k=10), A062308 (k=11).

Cf. A006530, A052539.

Programs

Magma

[Maximum(PrimeDivisors(4^n+1)): n in [0..35]];

Mathematica

Table[FactorInteger[4^n + 1][[-1, 1]], {n, 0, 30}]

PARI

a(n)=my(f=factor(4^n+1)[,1]); f[#f] \\ Charles R Greathouse IV, Jul 12 2016

Formula

a(n) = A006530(A052539(n)). - Michel Marcus, Jul 11 2016

a(2n) = A002590(n). a(2n+1) = A229747(n). - R. J. Mathar, Feb 28 2018

a(n) = A002587(2*n). - Amiram Eldar, Feb 01 2020

Extensions

Terms to a(100) in b-file from Vincenzo Librandi, Jul 12 2016

a(101)-a(531) in b-file from Amiram Eldar, Feb 01 2020

a(532)-a(583) in b-file from Max Alekseyev, Apr 25 2022, Mar 15 2025