A001166 -id:A001166 - 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

A045494 Smallest positive integer requiring at least n letters (not including hyphens) when spelled out in English.

Original entry on oeis.org

1, 1, 1, 3, 3, 11, 13, 13, 17, 23, 23, 73, 101, 103, 103, 111, 113, 113, 117, 123, 123, 173, 323, 373, 1103, 1103, 1111, 1113, 1113, 1117, 1123, 1123, 1173, 1323, 1373, 3323, 3373, 11373, 13323, 13373, 17373, 23323, 23373, 73373, 101373, 103323

Offset: 1

Brian Galebach

Assumes British definition of billion, trillion. Also assumes no 'and' is used to spell integers.

Cf. A000916, A001166, A014388, A045494, A045495, A045495, A167509.

Added "at least" in definition M. F. Hasler, Nov 18 2009

A014388 a(2n-1) = n+2, a(2n) = smallest number requiring n+2 letters in English.

Original entry on oeis.org

3, 1, 4, 4, 5, 3, 6, 11, 7, 15, 8, 13, 9, 17, 10, 24, 11, 23, 12, 73, 13, 3000, 14, 11000, 15, 15000, 16, 101, 17, 104, 18, 103, 19, 111, 20, 115, 21, 113, 22, 117, 23, 124, 24, 123, 25, 173, 26, 323, 27, 373, 28, 1104, 29, 1103, 30, 1111, 31

Offset: 1

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Uses number forms containing "and"; that is, "one hundred and one" rather than "one hundred one". - Sean A. Irvine, Oct 20 2018

Cf. A000916, A001166, A014388, A045494, A045495.

More terms from Sean A. Irvine, Oct 20 2018

A045495 Smallest positive integer requiring at least n letters (not including hyphens) to be spelled out in English.

Original entry on oeis.org

1, 1, 1, 3, 3, 11, 13, 13, 17, 23, 23, 73, 101, 101, 101, 101, 103, 103, 111, 113, 113, 117, 123, 123, 173, 323, 373, 1103, 1103, 1111, 1113, 1113, 1117, 1123, 1123, 1173, 1323, 1373, 3323, 3373, 11373, 13323, 13373, 17373, 23323, 23373, 73373, 101123

Offset: 1

Brian Galebach

Assumes British definition of billion trillion. Also assumes 'and' is used in numbers such as 'one hundred and one'.

Cf. A000916, A001166, A014388, A045494, A045495, A167509.

Added "at least" in definition, and a cross-reference M. F. Hasler, Nov 18 2009

A000916 a(2n) = n+2, a(2n-1) = smallest number requiring n+2 letters in English.

Original entry on oeis.org

1, 3, 4, 4, 3, 5, 11, 6, 15, 7, 13, 8, 17, 9, 24, 10, 23, 11, 73, 12, 3000, 13, 11000, 14, 15000, 15, 101, 16, 104, 17, 103, 18, 111, 19, 115, 20, 113, 21, 117, 22, 124, 23, 123, 24, 173, 25, 323, 26, 373, 27, 1104, 28, 1103, 29, 1111, 30

Offset: 1

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Requires presence of "and" at appropriate place, e.g. "one hundred and one". - Sean A. Irvine, Aug 29 2011.

Cf. A001166, A014388, A045494, A045495.

More terms from Sean A. Irvine, Aug 28 2011

A052193 Positive integers sorted by length of name in American English and alphabetical order.

Original entry on oeis.org

1, 6, 10, 2, 5, 4, 9, 8, 50, 40, 7, 60, 3, 80, 11, 90, 30, 12, 20, 15, 70, 16, 18, 51, 56, 52, 41, 46, 42, 14, 19, 61, 66, 62, 13, 81, 86, 82, 55, 54, 59, 45, 44, 49, 91, 96, 92, 17, 65, 64, 69, 31, 36, 32, 21, 26, 22, 85, 84, 89, 58, 57, 53, 48, 47, 43, 95, 94, 99

Offset: 1

Henry Bottomley, Jan 28 2000

Using U.S. nomenclature. Limited to numbers < 10^66.
If allowing larger numbers, one googol (10^100) would occur after 92.
The final term is a(10^66-1) = 373373373373373373373373373373373373373373373373373373373373373373.

			The three-letter numbers in alphabetical order: one, six, ten, two.

Hans Havermann, Table of n, a(n) for n = 1..1296

Cf. A001166, A052194, A121064.

More terms from Larry Reeves (larryr(AT)acm.org), Oct 02 2000

A052194 Positive integers sorted by length of name in English and numerical order.

Original entry on oeis.org

1, 2, 6, 10, 4, 5, 9, 3, 7, 8, 40, 50, 60, 11, 12, 20, 30, 80, 90, 15, 16, 70, 13, 14, 18, 19, 41, 42, 46, 51, 52, 56, 61, 62, 66, 17, 21, 22, 26, 31, 32, 36, 44, 45, 49, 54, 55, 59, 64, 65, 69, 81, 82, 86, 91, 92, 96, 24, 25, 29, 34, 35, 39, 43, 47, 48, 53, 57, 58, 63, 67, 68

Offset: 0

Henry Bottomley, Jan 28 2000

Using U.S. nomenclature. Limited values to numbers < 10^36. If allowing greater values, one googol (10^100) would come after 96.

0 could be inserted before '11', as 0's alternative name is 'nought'. - Dylan Nicholson (wizofaus(AT)hotmail.com), May 08 2006

			The three-letter numbers are one, two, six and ten.

Cf. A001166, A052193.

More terms from Larry Reeves (larryr(AT)acm.org), Oct 02 2000

A052196 Largest natural number less than 10^66 requiring exactly n letters in English.

Original entry on oeis.org

10, 9, 60, 90, 70, 66, 96, 10000000000, 10000000000000, 10000000000000000000000000000000000, 10000000000000000000000000, 10000000000000000000000000000000000000, 10000000000000000000000000000000000000000000000000000000000000000, 9000000000000000000000000000000000000000000000000000000000000000

Offset: 3

Henry Bottomley, Jan 28 2000

This uses US nomenclature: no conjunctive 'and'; 10^10 = 'ten billion'.
This is the 'largest' counterpart to A080777, which gives the smallest positive integer with exactly n letters.
Because of the definition's size limitation, a(758) will be the largest term in this finite sequence; a(758) = 878878878878878878878878878878878878878878878878878878878878878878.

			The largest numbers (<10^66) using 10 to 15 letters:
10: 10*10^9 = ten billion
11: 10*10^12 = ten trillion
12: 10*10^33 = ten decillion
13: 10*10^24 = ten septillion
14: 10*10^36 = ten undecillion
15: 10*10^63 = ten vigintillion

Hans Havermann, Table of n, a(n) for n = 3..758
Hans Havermann, Growth illustration for this sequence
Hans Havermann, Define, divide, and conquer
Pegg, E. Jr. and Weisstein, E. W. Mathematica's Google Aptitude: problem #20 MathWorld Headline news, Oct 13, 2004.

Cf. A001166, A080777.

k=100;lst=StringLength/@StringReplace[IntegerName/@Range[k],
{"-"-> ""," "-> ""}];max[n_]:=Last[Position[lst,n]];
max/@Range[3,9]//Flatten (* Ivan N. Ianakiev, Oct 07 2015 *)

a(11) from Brian Galebach, Feb 06 2004

Edited and extended by Hans Havermann, Nov 08 2013

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

cp's OEIS Frontend

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

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A045494 Smallest positive integer requiring at least n letters (not including hyphens) when spelled out in English.

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A014388 a(2n-1) = n+2, a(2n) = smallest number requiring n+2 letters in English.

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A045495 Smallest positive integer requiring at least n letters (not including hyphens) to be spelled out in English.

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A000916 a(2n) = n+2, a(2n-1) = smallest number requiring n+2 letters in English.

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A052193 Positive integers sorted by length of name in American English and alphabetical order.

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A052194 Positive integers sorted by length of name in English and numerical order.

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A052196 Largest natural number less than 10^66 requiring exactly n letters in English.

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

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