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A129774
Main diagonal of table of length of English names of numbers.
2
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
OFFSET
1,2
COMMENTS
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}
From Michael S. Branicky, Jul 13 2020: (Start)
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)
FORMULA
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.
EXAMPLE
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}.
PROG
(Python)
def A129774(n):
i, found, limit = 0, 0, 10**2
while found < n-2 and i < limit:
i += 1
found += 1*(A005589(i)==n)
return i*(i < limit)
print([A129774(i) for i in range(3, 12)]) # Michael S. Branicky, Jul 13 2020
KEYWORD
easy,fini,nonn,word,less
AUTHOR
Jonathan Vos Post, May 17 2007, May 21 2007
EXTENSIONS
Corrected and edited by Danny Rorabaugh, May 13 2016
Corrected terms a(10)-a(18) and table in comments from 9; added terms from a(20) - Michael S. Branicky, Jul 13 2020
STATUS
approved