A185121 Smallest prime factor of 10^(2^n) + 1.

11, 101, 73, 17, 353, 19841, 1265011073, 257, 10753, 1514497, 1856104284667693057, 106907803649, 458924033, 3635898263938497962802538435084289

Offset

0,1

Comments

10^k+1 can only be prime if k is a power of 2. So far the only known primes of this form are a(0) = 11 and a(1) = 101. [Edited by M. F. Hasler, Aug 03 2019]

a(n) >= 2^(n+1)+1; we have a(n) = 2^(n+1)+1 for n=3, n=7, and n=15.

a(14) > 10^16. - Max Alekseyev, Jun 28 2013

From the Keller link a(15)-a(20) = 65537, 8257537, 175636481, 639631361, 70254593, 167772161. - Ray Chandler, Dec 27 2013

Links

Table of n, a(n) for n=0..13.

FactorDB, Factorizations of 10^(2^n)+1

Makoto Kamada, Factorizations of 100...001

Wilfrid Keller, Prime factors of generalized Fermat numbers Fm(10) and complete factoring status

Formula

a(n) = A038371(2^n). - M. F. Hasler, Jul 30 2019

Example

For n=2, a(2)=73 since 10^(2^2) + 1 = 10001 = 73 * 137.

Mathematica

Table[With[{k = 2^n}, FactorInteger[10^k + 1]][[1, 1]], {n, 0, 13, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

Prog

(PARI) a(n) = factor(10^(2^n)+1)[1, 1] \\ Michel Marcus, May 30 2013

Crossrefs

Cf. A038371, A000533, A000215, A093179, A102050

Essentially the same as A102050. - Sean A. Irvine, Feb 17 2013

Sequence in context: A330290 A330291 A318647 * A133835 A326108 A292451

Adjacent sequences: A185118 A185119 A185120 * A185122 A185123 A185124

Keyword

nonn,more,hard

Author

Sergio Pimentel, Jan 22 2012

Status

approved