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%I #42 Jan 05 2025 19:51:39
%S 38,381,3811,38111,381111,3811111,38111111,381111111,3811111111,
%T 38111111111,381111111111,3811111111111,38111111111111,
%U 381111111111111,3811111111111111,38111111111111111,381111111111111111,3811111111111111111,38111111111111111111,381111111111111111111
%N 38 followed by n ones.
%C Jones proves that 38 is the smallest composite number that produces only composite numbers when 1 is repeatedly appended to it.
%C Stan Wagon, who is working on similar numbers ending in 3, 7, and 9, reports that the upper bounds for those numbers are 4070, 891, and 10175, respectively. - _T. D. Noe_, Mar 08 2011
%C Today Stan Wagon found a 28899-digit probable prime 851777...777, which eliminated 851 as a possible prefix for the 7-case. Because it is easy to show the 891777...777 is composite for any number of 7's, this means that 891 is the solution for the 7-case of this problem. - _T. D. Noe_, Mar 08 2011
%H Vincenzo Librandi, <a href="/A187078/b187078.txt">Table of n, a(n) for n = 0..100</a>
%H Lenny Jones, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.153">When does appending the same digit repeatedly on the right of a positive integer generate a sequence of composite integers?</a>, Amer. Math. Monthly, 118 (2011), 153-160.
%H Lenny Jones and Maria Markovich, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-2/JonesMarkovich.pdf">Generating Composite Sequences by Appending Digits to Special Types of Integers</a>, Fibonacci Quart. 52 (2014), no. 2, 148-159.
%H Michael Penn, <a href="https://www.youtube.com/watch?v=v1x-1zKjiyA">From the Benelux Math Olympiad.</a>, YouTube video, 2021.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (11,-10).
%F From _Bruno Berselli_, Mar 09 2011: (Start)
%F G.f.: (38 - 37*x)/((1 - x)*(1 - 10*x)).
%F a(n) = (343*10^n - 1)/9. (End)
%F a(n) = 10*a(n-1) + 1 with n>0, a(0)=38. - _Vincenzo Librandi_, Jun 07 2011
%t t = {38}; Do[AppendTo[t, 10*t[[-1]] + 1], {19}]; t
%t NestList[10#+1&,38,20] (* _Harvey P. Dale_, Sep 20 2017 *)
%o (Magma) [Seqint([1: n in [1..k]] cat Intseq(38)): k in [0..19]]; // _Bruno Berselli_, Mar 09 2011
%o (PARI) a(n)=343*10^n\9 \\ _Charles R Greathouse IV_, Sep 24 2012
%K nonn,base,easy
%O 0,1
%A _T. D. Noe_, Mar 03 2011