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A062397 a(n) = 10^n + 1.

%I A062397 #66 Mar 18 2025 15:04:03
%S A062397 2,11,101,1001,10001,100001,1000001,10000001,100000001,1000000001,
%T A062397 10000000001,100000000001,1000000000001,10000000000001,
%U A062397 100000000000001,1000000000000001,10000000000000001,100000000000000001
%N A062397 a(n) = 10^n + 1.
%C A062397 The first three terms (indices 0, 1 and 2) are the only known primes. Moreover, the terms not of the form a(2^k) are all composite, except for a(0). Indeed, for all n >= 0, a(2n+1) is divisible by 11, a(4n+2) is divisible by 101, a(8n+4) is divisible by 73, a(16n+8) is divisible by 17, a(32n+16) is divisible by 353, a(64n+32) is divisible by 19841, etc. - _M. F. Hasler_, Nov 03 2018 [Edited based on the comment by _Jeppe Stig Nielsen_, Oct 17 2019]
%C A062397 This sequence also results when each term is generated by converting the previous term into a Roman numeral, then replacing each letter with its corresponding decimal value, provided that the vinculum is used and numerals are written in a specific way for integers greater than 3999, e.g., IV with a vinculum over the I and V for 4000. - _Jamie Robert Creasey_, Apr 14 2021
%C A062397 By Mihăilescu's theorem, a(n) can never be a perfect power (see "Catalan's conjecture" in Links). - _Marco Ripà_, Mar 10 2025
%H A062397 Vincenzo Librandi, <a href="/A062397/b062397.txt">Table of n, a(n) for n = 0..200</a>
%H A062397 Wikipedia, <a href="https://en.wikipedia.org/wiki/Catalan%27s_conjecture">Catalan's conjecture</a>.
%H A062397 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).
%F A062397 a(n) = 10*a(n-1) - 9 = A011557(n) + 1 = A002283(n) + 2.
%F A062397 From _Mohammad K. Azarian_, Jan 02 2009: (Start)
%F A062397 G.f.: 1/(1-x) + 1/(1-10*x).
%F A062397 E.g.f.: exp(x) + exp(10*x). (End)
%t A062397 LinearRecurrence[{11, -10},{2, 11},18] (* _Ray Chandler_, Aug 26 2015 *)
%t A062397 10^Range[0,20]+1 (* _Harvey P. Dale_, Jan 21 2020 *)
%o A062397 (Magma) [10^n + 1: n in [0..35]]; // _Vincenzo Librandi_, Apr 30 2011
%o A062397 (PARI) a(n)=10^n+1 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A062397 Except for the initial term, essentially the same as A000533. Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081 for numbers one more than powers, i.e., this sequence translated from base n (> 2) to base 10.
%Y A062397 Cf. A002283, A011557.
%Y A062397 Cf. A038371 (smallest prime factor), A185121.
%K A062397 easy,nonn
%O A062397 0,1
%A A062397 _Henry Bottomley_, Jun 22 2001