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A129774
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Main diagonal of table of length of English names of numbers.
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2
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1, 5, 8, 30, 0, 42, 36, 47, 79, 3000000, 606, 502, 301, 305, 420, 218, 181, 176, 233, 367, 578, 2101, 2105, 1607, 1540, 1616, 1311, 1232, 1235, 1298, 1423, 1787, 3348, 3793, 11375, 13358, 13823, 17577, 23339, 23833, 37777, 101398, 103384, 103875, 111478, 113394
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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a(n) is the n-th smallest positive integer with the property that, when spelled out in American English, has n+2 letters (or 0 if fewer than n such numbers exists).
The sequence is labeled "finite" because there is no widely accepted naming convention for arbitrarily large numbers.
The table {and length of each row} begins:
3..|.1..2..6.10.........{4}
4..|.4..5..9............{3}
5..|.3..7..8.40.50.60...{6}
6..|11.12.20.30.80.90...{6}
7..|15.16.70............{3}
8..|13.14.18.19.41.42.46.51.52.56.61.62.66.{13}
9..|17.21.22.26.31.32.36.44.45.49.54.55.59.64.65.69.81.82.86.91.92.96.{22}
10.|24.25.29.34.35.39.43.47.48.53.57.58.63.67.68.71.72.76.84.85.89.94.95.99...
11.|23.27.28.33.37.38.74.75.79.83.87.88.93.97.98.400.500.900.1000.2000.6000.10000.400000.5000000...
12.|73.77.78.300.700.800.4000.5000.9000.3000000.7000000.8000000.40000000.50000000.60000000...
13.|101.102.106.110.201.202.206.210.601.602.606.610.3000.700.8000.40000.50000.60000.1000001.1000002...
14.|104.105.109.204.205.209.401.402.406.410.501.502.506.510.604.605.609.901.902.906.910.1001.1002.1006...
15.|103.107.108.140.150.160.203.207.208.240.250.260.301.302.306.310.404.405.409.504.505.509.603.607...
16.|111.112.120.130.180.190.211.212.220.230.280.290.304.305.309.403.407.408.440.450.460.503.507.508...
17.|115.116.170.215.216.270.303.307.308.340.350.360.411.412.420.430.480.490.511.512.520.530.580.590...
18.|113.114.118.119.141.142.146.151.152.156.161.162.166.213.214.218.219.241.242.246.251.252.256.261...
19.|117.121.122.126.131.132.136.144.145.149.154.155.159.164.165.169.181.182.186.191.192.196.217.221...
20.|124.125.129.134.135.139.143.147.148.153.157.158.163.167.168.171.172.176.184.185.189.194.195.199...
21.|123.127.128.133.137.138.174.175.179.183.187.188.193.197.198.223.227.228.233.237.238.274.275.279...
22.|173.177.178.273.277.278.324.325.329.334.335.339.343.347.348.353.357.358.363.367.368.371.372.376...
23.|323.327.328.333.337.338.374.375.379.383.387.388.393.397.398.473.477.478.573.577.578.723.727.728..(End)
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LINKS
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FORMULA
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a(n) = A(n+2,n) where A(k,n) = n-th positive integer requiring exactly k letters (not including "and" or hyphens) in its English name, or 0 if no such integer.
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EXAMPLE
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a(1) = 1 because "one" is the first positive integer with 3 letters in its name.
a(2) = 5 because "five" is the second positive integer with 4 letters.
a(3) = 8 because "eight" is the third positive integer with 5 letters.
a(4) = 30 because "thirty" is the fourth positive integer with 6 letters.
a(5) = 0 because there are only three 7-letter positive integers: {15, 16, 70}.
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PROG
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(Python)
i, found, limit = 0, 0, 10**2
while found < n-2 and i < limit:
i += 1
return i*(i < limit)
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CROSSREFS
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KEYWORD
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easy,fini,nonn,word,less
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AUTHOR
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EXTENSIONS
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Corrected terms a(10)-a(18) and table in comments from 9; added terms from a(20) - Michael S. Branicky, Jul 13 2020
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STATUS
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approved
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