A185121 - OEIS

%I #58 Aug 15 2023 13:26:53

%S 11,101,73,17,353,19841,1265011073,257,10753,1514497,

%T 1856104284667693057,106907803649,458924033,

%U 3635898263938497962802538435084289

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

%C 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]

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

%C a(14) > 10^16. - _Max Alekseyev_, Jun 28 2013

%C From the Keller link a(15)-a(20) = 65537, 8257537, 175636481, 639631361, 70254593, 167772161. - _Ray Chandler_, Dec 27 2013

%H FactorDB, <a href="http://factordb.com/index.php?query=10^(2^n)%2B1&amp;use=n&amp;n=1">Factorizations of 10^(2^n)+1</a>

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit/10001.htm">Factorizations of 100...001</a>

%H Wilfrid Keller, <a href="http://web.archive.org/web/20160617002823/http://www.prothsearch.net/GFN10.html">Prime factors of generalized Fermat numbers Fm(10) and complete factoring status</a>

%F a(n) = A038371(2^n). - _M. F. Hasler_, Jul 30 2019

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

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

%o (PARI) a(n) = factor(10^(2^n)+1)[1, 1] \\ _Michel Marcus_, May 30 2013

%Y Cf. A038371, A000533, A000215, A093179, A102050

%Y Essentially the same as A102050. - _Sean A. Irvine_, Feb 17 2013

%K nonn,more,hard

%O 0,1

%A _Sergio Pimentel_, Jan 22 2012