A080777 -id:A080777 - cp's OEIS frontend

A001166 Smallest natural number requiring n letters in English.

1, 4, 3, 11, 15, 13, 17, 24, 23, 73, 3000, 11000, 15000, 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, 101123, 101173, 101323, 101373, 103323, 103373, 111373, 113323, 113373, 117373

Offset: 3

N. J. A. Sloane

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

			For n = 6, the smallest natural number requiring 6 letters in English is "eleven." - _Julia Carrigan_, Jan 19 2024

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Illustration of terms a(3) through a(75)

Cf. A000916, A014388, A045494, A045495, A080777.

Corrected and extended by Henry Bottomley, Jan 28 2000
Further corrected and extended by Brian Galebach, Feb 06 2004
Further corrected and illustration of terms by Sean A. Irvine, Mar 12 2012

A134629 Least nonnegative integer which requires n letters to spell in English, excluding spaces and hyphens. A right inverse of A005589.

Original entry on oeis.org

1, 0, 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, 113323, 113373, 117373, 123323, 123373

Offset: 3

Robert G. Wilson v, Nov 04 2007

Variant of A080777. - R. J. Mathar, Dec 13 2008

This is one of many possible right inverses of A005589, i.e., A005589 o A134629 = id (of course on the domain of this sequence, [3 .. oo)). It does not satisfy A134629 o A005589 = id. - M. F. Hasler, Feb 25 2018

			a(3) = 1: "one", a(4) = 0: "zero", a(5) = 3: "three", a(6) = 11: "eleven", a(7) = 15: "fifteen", etc.

Michael S. Branicky, Table of n, a(n) for n = 3..608 (terms < 10^54)
Hans Havermann, Table giving n and American English name for n, for 0 <= n <= 100999, without spaces or hyphens
Landon Curt Noll, The English Name of a Number.
Eric Weisstein's World of Mathematics, Number.

Cf. A005589, A052363, A080777.

(* first load Hans Havermann's text file a001477.txt and then *)
f[n_] := Length@ Characters@ ToString@ lst[[n]]; g[n_] :=  Block[{k = 1}, While[ f[k] != n, k++]; k]; Array[g, 41, 3] - 1 (* Robert G. Wilson v, May 26 2013 *)
f[n_] := Length@ StringPartition[ StringReplace[ IntegerName[n, "Words"], ", " | " " | "\[Hyphen]" -> ""], 1] (* after Giovanni Resta in A005589 *); t[] := 0; k = 1; While[k < 174000, a = f@k; If[t[a] == 0, t[a] = k]; k++]; t[4] = 0 (* only {0, 4, 5 & 9} have just four letters *); t@# & /@ Range[3, 54] (* _Robert G. Wilson v, May 25 2018 *)

A134629(n)=for(k=1,oo,A005589(k)==n&&return(k)) \\ M. F. Hasler, Feb 25 2018

from num2words import num2words as n2w
from itertools import count, islice
def f(n): return sum(1 for c in n2w(n).replace(" and", "") if c.isalpha())
def agen(): # generator of terms
    adict, n = dict(), 3
    for k in count(0):
        v = f(k)
        if v not in adict: adict[v] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 40))) # Michael S. Branicky, Feb 19 2023

More terms and reworded name from M. F. Hasler, Feb 25 2018

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

A084390 a(n) is the smallest positive integer > a(n-1) with exactly n letters when spelled in English.

Original entry on oeis.org

1, 4, 7, 11, 15, 18, 21, 24, 27, 73, 101, 104, 107, 111, 115, 118, 121, 124, 127, 173, 323, 373, 1104, 1107, 1111, 1115, 1118, 1121, 1124, 1127, 1173, 1323, 1373, 3323, 3373, 11373, 13323, 13373, 17373, 23323, 23373, 73373, 101373, 103323, 103373, 111373

Offset: 3

James Ong (blackshadowshade(AT)yahoo.com.au), Jun 27 2003

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

			a(5) = 7 because 'seven' has 5 letters.

Andrew Bolt et al., Maths fun, rec.puzzles group.
jaysmith et al., Number sequence - spot the pattern, rec.puzzles group.
Wikipedia, English numerals

Cf. A080777.

More terms from David Wasserman, Dec 20 2004

A167509 Least positive integer written with n different letters when spelled out in French.

Original entry on oeis.org

1, 6, 2, 3, 4, 17, 14, 22, 24, 53, 74, 92, 97

Offset: 2

M. F. Hasler, Nov 18 2009

There is no number which can be written in French using only one letter, therefore the sequence starts at offset n=2, cf. examples.
A variant of the definition would be the "least nonnegative integer ....", in which case a(4)=0 ("zéro" with "accent aigu" on the "e"), all other terms remaining the same.
It appears that letters "j", "k" and "w" don't occur in any number, while "m" and "l" first occur in "mille" (=1000), and "b" first occurs in "billion".
If an "é" with accent (as it occurs in "décillion") is considered as different from "e" without accent, the sequence should have 26-3+1 terms.

			The terms a(2),...a(14) correspond to the French words un, six, deux, trois, quatre, dix-sept, quatorze, vingt-deux, vingt-quatre, cinquante-trois, soixante-quatorze, quatre-vingt-douze, quatre-vingt-dix-sept.
Here, "vingt-quatre" is the first term which contains a letter occurring twice, and therefore has a length greater than n; we conjecture that this is the case for all subsequent terms.

Wiktionnaire, Annexe:Nombres de 1 à 100 en français (as of Nov 18, 2009).

Cf. A167507, A080777, A134629.

A167509(n) = { for( k=1,#A167508, A167508[k]==n & return(k)); error("Found no result for n="n) }

a(n) = min { k | A167508(k) = n }

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

A161353 Smallest natural number requiring n letters in Spanish.

Original entry on oeis.org

1, 3, 5, 4, 14, 40, 16, 17, 25, 24, 35, 34, 44, 54, 125, 124, 135, 134, 144, 154, 235, 234, 244, 254, 354, 444, 454, 1354, 1444, 1454, 2354, 2444, 2454, 3454, 4444, 4454, 14454, 16444, 16454, 17454, 24444, 24454, 34444, 34454, 44454, 54454, 124444, 124454

Offset: 3

Claudio Meller, Jun 07 2009

			1=uno (3 letters), 3=tres (4 letters), 5=cinco (5 letters), 4=cuatro (6 letters), 14= catorce (7 letters), 40=cuarenta (8 letters), 16=dieciséis (9 letters), 17= diecisiete (10 letters), etc.

Álvar Ibeas, Table of n, a(n) for n = 3..100
Wikipedia, Anexo:Nombres de los números en español

Cf. A080777 (English).

Offset corrected to 3 by Michel Marcus, Jul 09 2015

a(30)-a(34) corrected by Álvar Ibeas, Sep 18 2020

A274177 Smallest number whose shortest possible name when spelled in Japanese in 'On' reading has exactly n mora.

Original entry on oeis.org

2, 0, 11, 21, 31

Offset: 1

Felix Fröhlich, Jun 12 2016

Smallest number k such that A261126(k) = n.

			2 (ni) has 1 mora.
0 (ze-ro, re-i) has 2 mora.
11 (ju-i-chi) has 3 mora.
21 (ni-ju-i-chi) has 4 mora.
31 (sa-n-ju-i-chi) has 5 mora.

B. Bullock, Convert western numbers to kanji.
B. Bullock, What are the Japanese numbers from 1 to 100?.

Cf. A080777, A261126.

There were errors in the mora-count for numbers greater than 99, as explained in the comments in A261126. - N. J. A. Sloane, Jul 01 2016

A362442 a(1) = 6; thereafter a(n) = smallest number with a(n-1) letters in American English.

Original entry on oeis.org

6, 11, 23, 323, 1103323373373373373373373373373

Offset: 1

N. J. A. Sloane, Apr 22 2023

a(5) should be findable, but a(6) will probably not be well defined.

See A362441 for a British English version.

			a(2) = 11 since "eleven" is the smallest number with 6 letters.
a(3) = 23 since "twenty three" is the smallest with 11 letters.

GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See pages 92 and 275.

Cf. A005589, A362123, A362441.

a(n) = A080777(a(n-1)) for n > 1. - Michael S. Branicky, Apr 22 2023

a(5) from Michael S. Branicky, Apr 22 2023 using A080777

A356738 Smallest positive integer whose American English name consists of n words.

Original entry on oeis.org

1, 21, 101, 121, 1101, 1121, 21121, 101121, 121121, 1101121, 1121121, 21121121, 101121121, 121121121, 1101121121, 1121121121, 21121121121, 121121121121, 1101121121121, 1121121121121, 21121121121121, 101121121121121, 121121121121121, 1101121121121121

Offset: 1

Ivan N. Ianakiev, Aug 25 2022

Cf. A080777.

name[n_]:=StringReplace[IntegerName[n,"Words"],{"-"->" ",", "->" "}];
nameLen[n_]:=WordCount[name[n]]; f[1]=1; f[n_]:=f[n]=Module[{k=f[n-1]},
While[nameLen[k]

a(n) == 1 (mod 20).

cp's OEIS Frontend

A001166 Smallest natural number requiring n letters in English.

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A134629 Least nonnegative integer which requires n letters to spell in English, excluding spaces and hyphens. A right inverse of A005589.

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

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A084390 a(n) is the smallest positive integer > a(n-1) with exactly n letters when spelled in English.

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A167509 Least positive integer written with n different letters when spelled out in French.

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

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A161353 Smallest natural number requiring n letters in Spanish.

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A274177 Smallest number whose shortest possible name when spelled in Japanese in 'On' reading has exactly n mora.

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