A084390 -id:A084390 - cp's OEIS frontend

A080777 a(n), when spelled in English, is the smallest positive integer with exactly n letters.

1, 4, 3, 11, 15, 13, 17, 24, 23, 73, 101, 104, 103, 111, 115, 113, 117, 124, 123, 173, 323, 373, 1104, 1103, 1111, 1115, 1113, 1117, 1124, 1123, 1173, 1323, 1373, 3323, 3373, 11373, 13323, 13373, 17373, 23323, 23373, 73373, 101373, 103323, 103373, 111373

Offset: 3

Peter Kolbus (peter(AT)kolbusfamily.com), Mar 11 2003

In this version 101 is written "one hundred one", etc.

This uses the conventions that "and" is never used and two-digit numbers are not used before "hundred". The sequence is labeled "finite" because there is no widely accepted naming convention for arbitrarily large numbers. - David Wasserman, Dec 20 2004

			The 3rd term has 5 letters; the smallest positive integer with this number of letters is 3 (three).

Hans Havermann, Table of n, a(n) for n = 3..758
Hans Havermann, Growth illustration for this sequence

Cf. A001166, A052196 (the 'largest' analog of this sequence), A084390.

(* Works for a(n) up to 10^k *)
k=5;name[n_]:=IntegerName[n,"Words"];
nameLen[n_]:=StringLength[StringReplace[name[n],{" "-> "","-"-> "",","-> ""}]];
max[n_]:=Max[nameLen/@Range[10^(n-1)+1,10^n]];max10toK=max/@Range[k];
pos[n_Integer/;n>2]:=Position[Sort[Append[max10toK,n]],n,1][[1,1]]-1;
a[n_Integer/;n>2&&n<(10^k)+1]:=Module[{l=10^pos[n]},While[nameLen[l]!=n,l++];l];
a/@Range[3,40] (* Ivan N. Ianakiev, Sep 05 2018 *)

Corrected by James Ong (blackshadowshade(AT)yahoo.com.au), Jun 27 2003
More terms from Brian Galebach, Feb 06 2004
Edited by David Wasserman, Dec 20 2004

A129774 Main diagonal of table of length of English names of numbers.

Original entry on oeis.org

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

Jonathan Vos Post, May 17 2007, May 21 2007

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:
.|.1..2..6.10.........{4}
.|.4..5..9............{3}
.|.3..7..8.40.50.60...{6}
.|11.12.20.30.80.90...{6}
.|15.16.70............{3}
.|13.14.18.19.41.42.46.51.52.56.61.62.66.{13}
From Michael S. Branicky, Jul 13 2020: (Start)
.|17.21.22.26.31.32.36.44.45.49.54.55.59.64.65.69.81.82.86.91.92.96.{22}
|24.25.29.34.35.39.43.47.48.53.57.58.63.67.68.71.72.76.84.85.89.94.95.99...
|23.27.28.33.37.38.74.75.79.83.87.88.93.97.98.400.500.900.1000.2000.6000.10000.400000.5000000...
|73.77.78.300.700.800.4000.5000.9000.3000000.7000000.8000000.40000000.50000000.60000000...
|101.102.106.110.201.202.206.210.601.602.606.610.3000.700.8000.40000.50000.60000.1000001.1000002...
|104.105.109.204.205.209.401.402.406.410.501.502.506.510.604.605.609.901.902.906.910.1001.1002.1006...
|103.107.108.140.150.160.203.207.208.240.250.260.301.302.306.310.404.405.409.504.505.509.603.607...
|111.112.120.130.180.190.211.212.220.230.280.290.304.305.309.403.407.408.440.450.460.503.507.508...
|115.116.170.215.216.270.303.307.308.340.350.360.411.412.420.430.480.490.511.512.520.530.580.590...
|113.114.118.119.141.142.146.151.152.156.161.162.166.213.214.218.219.241.242.246.251.252.256.261...
|117.121.122.126.131.132.136.144.145.149.154.155.159.164.165.169.181.182.186.191.192.196.217.221...
|124.125.129.134.135.139.143.147.148.153.157.158.163.167.168.171.172.176.184.185.189.194.195.199...
|123.127.128.133.137.138.174.175.179.183.187.188.193.197.198.223.227.228.233.237.238.274.275.279...
|173.177.178.273.277.278.324.325.329.334.335.339.343.347.348.353.357.358.363.367.368.371.372.376...
|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)

			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}.

Cf. A000916, A001166, A014388, A045494, A045495, A005589, A080777, A084390.

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

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.

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

A259921 a(n) is the smallest positive integer > a(n-1) with exactly n letters when written in Spanish.

Original entry on oeis.org

1, 3, 5, 15, 30, 40, 50, 103, 105, 115, 130, 140, 150, 204, 214, 240, 250, 316, 317, 325

Offset: 3

Rodolfo Kurchan, Jul 08 2015

a(6) = 15 = "quince" (6 letters); 4 = "cuatro" also has 6 letters but is smaller than 5 = a(5) and this sequence is strictly increasing.

Spaces are not counted, so, e.g., "ciento tres" and "doscientos cuatro" are treated as a 10-letter word and a 16-letter word, respectively.

			a( 3) =   1 for the  3-letter word "uno";
a( 4) =   3 for the  4-letter word "tres";
a( 5) =   5 for the  5-letter word "cinco";
a( 6) =  15 for the  6-letter word "quince";
a( 7) =  30 for the  7-letter word "treinta";
a( 8) =  40 for the  8-letter word "cuarenta";
a( 9) =  50 for the  9-letter word "cincuenta";
a(10) = 103 for the 10-letter word "cientotres";
...
a(16) = 204 for the 16-letter word "doscientoscuatro";
....

Wikipedia, Anexo:Nombres de los números en español

Cf. A084390 (this sequence for English), A161353 (this sequence without the increasing condition).

cp's OEIS Frontend

A080777 a(n), when spelled in English, is the smallest positive integer with exactly n letters.

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A129774 Main diagonal of table of length of English names of numbers.

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A259921 a(n) is the smallest positive integer > a(n-1) with exactly n letters when written in Spanish.

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