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A349960
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Values taken by the function A067095 in the order of their appearance.
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2
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2, 1, 18, 181, 1817, 18175, 181757, 1817571, 18175719, 181757197, 1817571972, 18175719727, 181757197277, 1817571972772, 18175719727727, 181757197277277, 1817571972772779, 18175719727727795, 181757197277277957, 1817571972772779572, 18175719727727795720, 181757197277277957202
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OFFSET
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1,1
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COMMENTS
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a(2) < a(1), but thereafter this function increases monotonically without limit (see Krusemeyer reference).
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.
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).
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.
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REFERENCES
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Mark I. Krusemeyer, George T. Gilbert and Loren C. Larson, A Mathematical Orchard, Problems and Solutions, MAA, 2012, Problem 87, pp. 159-161.
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LINKS
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FORMULA
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EXAMPLE
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Floor(A019520(5)/A019519(5)) = floor(246810/13579) = floor(18.175859...) = 18, hence, 18 is a term.
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MATHEMATICA
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terms=5; f[i_]:=FromDigits@Flatten[IntegerDigits/@i];
k[q_]:=f[Range[2, 2q, 2]]/f[Range[1, 2q, 2]];
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PROG
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(Python)
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
from itertools import count
def A349960(n): # a more efficient implementation
if n <= 2:
return 3-n
a, b = '', ''
for i in count(1, 2):
a += str(i)
b += str(i+1)
ai, bi = int(a), int(b)
if len(a)+n-2 == len(b): return bi//ai
m = 10**(n-2-len(b)+len(a))
lb = bi*m//(ai+1)
ub = (bi+1)*m//ai
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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