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A328683 Positive integers that are equal to 99...99 (repdigit with n digits 9) times the sum of their digits.

%I A328683 #43 Feb 02 2025 17:12:17
%S A328683 81,1782,26973,359964,4499955,53999946,629999937,7199999928,
%T A328683 80999999919,899999999910,9899999999901,107999999999892,
%U A328683 1169999999999883,12599999999999874,134999999999999865,1439999999999999856,15299999999999999847,161999999999999999838
%N A328683 Positive integers that are equal to 99...99 (repdigit with n digits 9) times the sum of their digits.
%C A328683 The idea of this sequence comes from a problem during the annual Moscow Mathematical Olympiad (MMO) in 2001 (see reference).
%D A328683 Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005, Level B, Problem 5, 2001, MSRI, 2011, p. 8 and 70/71.
%H A328683 Colin Barker, <a href="/A328683/b328683.txt">Table of n, a(n) for n = 1..995</a>
%H A328683 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%H A328683 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (22,-141,220,-100).
%F A328683 a(n) = 9 * n * (10^n - 1).
%F A328683 From _Colin Barker_, Feb 25 2020: (Start)
%F A328683 G.f.: 81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2).
%F A328683 a(n) = 22*a(n-1) - 141*a(n-2) + 220*a(n-3) - 100*a(n-4) for n>4.
%F A328683 (End)
%F A328683 From _Michel Marcus_, Feb 25 2020: (Start)
%F A328683 a(n) = 9*A110807(n).
%F A328683 a(n) = n*A086580(n). (End)
%e A328683 359964 = 36 * 9999 and the digital sum of 359964 = 36 , so 359964 = a(4).
%p A328683 C:=seq(9*n*(10^n-1),n=1..20);
%t A328683 Table[9*n*(10^n - 1), {n, 1, 18}] (* _Amiram Eldar_, Feb 25 2020 *)
%t A328683 LinearRecurrence[{22,-141,220,-100},{81,1782,26973,359964},20] (* _Harvey P. Dale_, Feb 02 2025 *)
%o A328683 (PARI) Vec(81*x*(1 - 10*x^2) / ((1 - x)^2*(1 - 10*x)^2) + O(x^20)) \\ _Colin Barker_, Feb 25 2020
%Y A328683 Cf. A057147, A086580, A110807.
%K A328683 nonn,base
%O A328683 1,1
%A A328683 _Bernard Schott_, Feb 25 2020