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A349960 Values taken by the function A067095 in the order of their appearance.

%I A349960 #75 Dec 13 2021 03:08:15
%S A349960 2,1,18,181,1817,18175,181757,1817571,18175719,181757197,1817571972,
%T A349960 18175719727,181757197277,1817571972772,18175719727727,
%U A349960 181757197277277,1817571972772779,18175719727727795,181757197277277957,1817571972772779572,18175719727727795720,181757197277277957202
%N A349960 Values taken by the function A067095 in the order of their appearance.
%C A349960 a(2) < a(1), but thereafter this function increases monotonically without limit (see Krusemeyer reference).
%C A349960 The record values > 2 of A067095(m) occur when m = 5, 50, 500, 5000, .... This happens precisely when the corresponding numerator A019520(m) goes from 2/4/6/8/10/12/....../999...98 to 2/4/6/8/10/12/....../999...98/1000...00, where here / means concatenation.
%C A349960 If a(n) is a k-digit number (k = A055642(a(n))), then 1.8 * 10^(k-1) < a(n) < 1.9 * 10^(k-1).
%C A349960 If we consider the sequence u(n) = a(n)/10^(k-1) where k = length(a(n)); we have u(n) is increasing with an upper bound 1.9; so, this sequence u(n) is convergent and, conjecture, this limit = 1.81757197277277957... found by _Giorgos Kalogeropoulos_; now, from this limit, it is possible to get the successive terms of this sequence here.
%D A349960 Mark I. Krusemeyer, George T. Gilbert and Loren C. Larson, A Mathematical Orchard, Problems and Solutions, MAA, 2012, Problem 87, pp. 159-161.
%H A349960 Chai Wah Wu, <a href="/A349960/b349960.txt">Table of n, a(n) for n = 1..1001</a>
%F A349960 a(n) = floor(k((n + 6)/2)*10^(n - 1 - ceiling(log_10(k((n + 6)/2))))) for k(n) = A019520(n)/A019519(n) and n >= 2 (conjectured). - _Giorgos Kalogeropoulos_, Dec 10 2021
%e A349960 Floor(A019520(5)/A019519(5)) = floor(246810/13579) = floor(18.175859...) = 18, hence, 18 is a term.
%t A349960 terms=5; f[i_]:=FromDigits@Flatten[IntegerDigits/@i];
%t A349960 k[q_]:=f[Range[2,2q,2]]/f[Range[1,2q,2]];
%t A349960 DeleteDuplicates@Table[Floor[k@n],{n,10^(terms-2)/2}] (* _Giorgos Kalogeropoulos_, Dec 10 2021 *)
%o A349960 (Python)
%o A349960 def A349960(n): return 3-n if n <= 2 else int("".join(str(d) for d in range(2,10**(n-2)+1,2)))//int("".join(str(d) for d in range(1,10**(n-2),2))) # _Chai Wah Wu_, Dec 10 2021
%o A349960 from itertools import count
%o A349960 def A349960(n): # a more efficient implementation
%o A349960     if n <= 2:
%o A349960         return 3-n
%o A349960     a, b = '', ''
%o A349960     for i in count(1,2):
%o A349960         a += str(i)
%o A349960         b += str(i+1)
%o A349960         ai, bi = int(a), int(b)
%o A349960         if len(a)+n-2 == len(b): return bi//ai
%o A349960         m = 10**(n-2-len(b)+len(a))
%o A349960         lb = bi*m//(ai+1)
%o A349960         ub = (bi+1)*m//ai
%o A349960         if lb == ub: return lb # _Chai Wah Wu_, Dec 10 2021
%Y A349960 Cf. A019519, A019520, A055642, A067095.
%K A349960 nonn,base
%O A349960 1,1
%A A349960 _Bernard Schott_, Dec 07 2021
%E A349960 a(5)-a(7) from _Michel Marcus_, Dec 07 2021
%E A349960 a(8)-a(9) from _Martin Ehrenstein_, Dec 10 2021
%E A349960 a(10)-a(22) from _Chai Wah Wu_, Dec 10 2021