A005589 -id:A005589 - cp's OEIS frontend

A060403 Each term is the previous term plus the number of letters in the previous number, as conventionally spelled out in American English.

1, 4, 8, 13, 21, 30, 36, 45, 54, 63, 73, 85, 95, 105, 119, 137, 158, 178, 200, 210, 223, 244, 263, 283, 304, 320, 338, 361, 381, 402, 416, 434, 455, 475, 497, 519, 538, 560, 576, 597, 619, 637, 658, 678, 700, 712, 730, 748, 770, 789, 811, 829, 851, 871, 893

Offset: 1

Kevin Langdon (kevin.langdon(AT)polymath-systems.com), Apr 05 2001

Increases more slowly than A160395 since American English does not use 'and' to separate hundreds from the rest of the number. E.g., 619 = "six hundred nineteen" in American English but "six hundred and nineteen" in British English. - Carl R. White, May 12 2009

			a(2)=4 because a(1)=1 and 4 is 1 plus the number of letters in "one," 3.

GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See pages 49 and 214.

Michael De Vlieger, Table of n, a(n) for n = 1..10001

Cf. A005589 See A139097 for another version.

For British English see A160395. - Carl R. White, May 12 2009

NestList[#+Length[Select[Characters[IntegerName[#,"Words"]],LetterQ ]]&,1,54] (* James C. McMahon, Jul 30 2024 *)

More terms from Carl R. White, May 12 2009

A006944 Number of letters in the n-th ordinal number (in American English).

Original entry on oeis.org

5, 6, 5, 6, 5, 5, 7, 6, 5, 5, 8, 7, 10, 10, 9, 9, 11, 10, 10, 9, 11, 12, 11, 12, 11, 11, 13, 12, 11, 9, 11, 12, 11, 12, 11, 11, 13, 12, 11, 8, 10, 11, 10, 11, 10, 10, 12, 11, 10, 8, 10, 11, 10, 11, 10, 10, 12, 11, 10, 8, 10, 11, 10, 11, 10, 10, 12, 11, 10, 10, 12, 13, 12, 13, 12, 12

Offset: 1

N. J. A. Sloane

a(0) is ambiguous (see Wikipedia: English numerals link). It is either 6 or 7 depending on whether the word used is 'zeroth' or 'noughth'. - Jon Perry, Nov 01 2014

The ordinal numbers 101st, 102nd, etc., are commonly spoken as "one hundred and first," "one hundred and second," etc., with the word "and" following the word "hundred." The more concise wordings "one hundred first," "one hundred second," etc. (without the word "and") are recommended by numerous authoritative reference works on American English, including the AP Style Guide and the U.S. Government Printing Office Style Manual. The American convention of omitting the "and" is followed in the b-file. - Jon E. Schoenfield, Nov 04 2014

			"First" has 5 letters, so a(1)=5.
Hyphens and spaces are not counted, so, e.g., a(21)=11 ("twenty-first") and a(100)=12 ("one hundredth").

Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file (Science Section).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
Associated Press, Stylebook
U.S. Government Printing Office, Style Manual 2008
Wikipedia, 101
Wikipedia, English numerals
Wikipedia, Ordinal numbers
Index entries for sequences related to number of letters in n

Cf. A005589.

Cf. A196278 (analog for French), A006969 (variant for French counting spaces and hyphens).

from num2words import num2words
def a(n): return sum(1 for c in num2words(n, to='ordinal').replace(" and", "") if c.isalpha())
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, Aug 08 2021 edited Jul 12 2022

More terms from Jon E. Schoenfield, Aug 13 2007

A007005 Number of characters in the French spelling of n, including spaces and hyphens.

Original entry on oeis.org

4, 2, 4, 5, 6, 4, 3, 4, 4, 4, 3, 4, 5, 6, 8, 6, 5, 8, 8, 8, 5, 11, 10, 11, 12, 10, 9, 10, 10, 10, 6, 12, 11, 12, 13, 11, 10, 11, 11, 11, 8, 14, 13, 14, 15, 13, 12, 13, 13, 13, 9, 15, 14, 15, 16, 14, 13, 14, 14, 14, 8, 14, 13, 14, 15, 13, 12, 13, 13, 13, 12, 16, 14, 15, 17, 15, 14

Offset: 0

Simon Plouffe

This sequence gives the number of characters, including spaces and hyphens, in the French spelling of the numbers; e.g., a(21) = 11 = #"vingt et un", a(22) = 10 = #"vingt-deux". - M. F. Hasler, Nov 18 2009
See A167507 for a variant where only letters are counted, but spaces and hyphens are not counted. - M. F. Hasler, Jun 03 2012
See A052360 for the English version (and A005589 for the letters-only variant); A007208 for the German version. - M. F. Hasler, Sep 20 2014
This refers to the official French spelling, Swiss or Belgian variants ("septante", ...) are not considered here. - M. F. Hasler, Sep 21 2014

			a(0) = 4 = #"zéro"
a(80) = 13 = #"quatre-vingts"
a(999) = 31 = #"neuf cent quatre-vingt-dix-neuf"
a(1000) = 5 = # "mille"

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

M. F. Hasler, Table of n, a(n) for n = 0..1000
Wiktionnaire, Annexe:Nombres de 1 à 100 en français. [From _M. F. Hasler_, Nov 18 2009]
Index entries for sequences related to number of letters in n

Cf. A005589, A007208, A052360, A167507.

A007005(n)=#French(n)   /* cf. A167507 for PARI code of "French()" - M. F. Hasler, Jun 03 2012 */

a(71) and a(74) corrected by M. F. Hasler, Jun 03 2012

Example completed by Rémy Sigrist, Sep 09 2018

A052362 Indices of records in length of English name of n including spaces and dashes (A052360): n such that k < n => A052360(k) < A052360(n).

Original entry on oeis.org

0, 3, 11, 13, 17, 21, 23, 73, 101, 103, 111, 113, 117, 121, 123, 173, 323, 373, 1101, 1103, 1111, 1113, 1117, 1121, 1123, 1173, 1323, 1373, 3323, 3373, 11373, 13323, 13373, 17373, 21373, 23323, 23373, 73373, 101323, 101373, 103323, 103373, 111373, 113323, 113373, 117373

Offset: 1

Allan C. Wechsler, Mar 07 2000

From M. F. Hasler, Aug 12 2020: (Start)
This sequence uses A052360 which counts all characters in the English name of the numbers, including spaces and hyphens, in contrast to A052363 which uses A005589 which only counts the letters. Thus, e.g., "twenty-one" is in this sequence but not in A052363.
It appears that from 1323 on, all terms end in -323 or in -373. After 117(373) these are prefixed by 121, 123, 173, 323, 373 (thousand). Then the next terms is 1'103'323, and after 1'373'373, the next terms are > 3*10^6, then > 11*10^6, etc.
I conjecture that from 103 on all d-digit terms have a smaller (most often but not always (d-1)-digit) term as suffix, and from 173 on they also have an earlier term as prefix. (End)

			From _M. F. Hasler_, Aug 12 2020: (Start)
The first term is zero, since all other nonnegative integers (thus certainly all those with longer names) are larger than zero.
"One" and "two" are not in the sequence, since "zero" is smaller but has a longer name.
"Three" is again in the sequence since all smaller numbers (0, 1 and 2) have shorter names. And so on. (End)

Cf. A005589, A052360, A052363.

m=0;for(n=0,2e5, if(m<A052360(n), m=A052360(n); print1(n", "))) \\ M. F. Hasler, Aug 12 2020

from itertools import count, islice
from num2words import num2words as n2w
def f(n): return len(n2w(n).replace(" and", "").replace(chr(44), ""))
def agen():
    record = 0
    for n in count(0):
        value = f(n)
        if value > record: yield n; record = value
print(list(islice(agen(), 46))) # Michael S. Branicky, Jul 12 2022

Minor edits by Ray Chandler, Jul 22 2009

Name and example edited and more terms by M. F. Hasler, Aug 12 2020

A073029 Names for numbers in American English, with each letter transformed into its index in the alphabet.

Original entry on oeis.org

26, 5, 18, 15, 15, 14, 5, 20, 23, 15, 20, 8, 18, 5, 5, 6, 15, 21, 18, 6, 9, 22, 5, 19, 9, 24, 19, 5, 22, 5, 14, 5, 9, 7, 8, 20, 14, 9, 14, 5, 20, 5, 14, 5, 12, 5, 22, 5, 14, 20, 23, 5, 12, 22, 5, 20, 8, 9, 18, 20, 5, 5, 14, 6, 15, 21, 18, 20, 5, 5, 14, 6, 9, 6, 20, 5, 5, 14, 19, 9, 24, 20, 5

Offset: 0

Zak Seidov, Aug 22 2002

Irregular triangle read by rows, in which row n lists the successive indices of the letters in the American English name for n. For example, row one is 15, 14, 5. - N. J. A. Sloane, Apr 22 2023
Find smallest n's for which a(n)=1,2,3,4,...,26.
A: The numbers 10 and 11 never occur. The rows in which the others occur first (assuming use of the "short scale") are 1000 (thousAnd), 10^9 (Billion), 10^27 (oCtillion), 100 (hunDred), 0 (zEro), 4 (Four), 8 (eiGht), 3 (tHree), 5 (fIve), --, -- (j & k don't occur in English names of numbers), 11 (eLeven), 10^6 (Million), 1 (oNe), 0 (zerO), 10^24 (sePtillion), 10^15 (Quadrillion), 0 (zeRo), 6 (Six), 2 (Two), 4 (foUr), 5 (fiVe), 2 (tWo), 6 (siX), 20 (twentY), 0 (Zero). Converting the position in the row plus the preceding row lengths to a linear index n this yields (after subtracting 1 to match offset 0 of the sequence): 18452, ?, ?, 864, 1, 15, 33, 11, 20, -, -, 44, ?, 5, 3, ?, ?, 2, 23, 7, 17, 21, 8, 25, 115, 0. The graph nicely shows the position & frequency of the individual letters. - M. F. Hasler, Feb 06 2016

			Top row is for "zero"; "z" is the 26th letter in the alphabet, "e" the fifth, "r" the 18th and "o" the 15th, so the first row is 26,5,18,15.

M. F. Hasler, Table of n, a(n) for n = 0..18454 (rows 0..1000 "flattened"), Feb 06 2016
M. F. Hasler, rows 0 - 1000, Feb 06 2016
Robert G. Wilson v, American English names for the numbers from 0 to 100999 without spaces or hyphens, Jun 08 2010

Cf. A005589 (row lengths).
Cf. A073327 (row sums).
Cf. A072922.

alef=Characters["abcdefghijklmnopqrstuvwxyz"]; numb="zeroonetwothreefourfivesixseveneightnineteneleven[...]"; ch=Characters[numb]; seq[n_] := Position[alef, ch[[n]]][[1, 1]] (* Franklin T. Adams-Watters, Jun 02 2006 *)
(* see the link to a001477.txt, copy the first 17 lines and then paste and assign to the variable 'str' as a string as follows *)
str = "zeroonetwothreefourfivesixseveneightnineteneleventwelvethirteenfourteenfifteensixteen";
Characters@ str /. Flatten[ Table[ {FromCharacterCode[96 + n] -> n}, {n, 26}]] (* Robert G. Wilson v, Jun 08 2010 *)

A073029_row(n)=select(t->t>64,Vec(Vecsmall(English(n))))%32 \\ See A052360 for English(). - M. F. Hasler, Feb 06 2016

from num2words import num2words
def row(n): return [ord(c)-96 for c in num2words(n).replace(" and", "") if c.isalpha()]
print([e for n in range(17) for e in row(n)]) # Michael S. Branicky, Apr 22 2023

Corrected and extended by Franklin T. Adams-Watters, Jun 02 2006, Oct 24 2006

A006090 Expansion of bracket function.

Original entry on oeis.org

1, -6, 21, -56, 126, -252, 463, -804, 1365, -2366, 4368, -8736, 18565, -40410, 87381, -184604, 379050, -758100, 1486675, -2884776, 5592405, -10919090, 21572460, -43144920, 87087001, -176565486, 357913941, -723002336

Offset: 0

David Callan

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2, issue 4, (1964), 241-260.
Problems Drive, Eureka, 37 (1974), 8-11, 32-33, 24-27. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (-6,-15,-20,-15,-6).

Column 6 of A307047.

Cf. A000748, A000749, A000750, A001659.

CoefficientList[Series[1/((1+x)^6-x^6),{x,0,30}],x] (* or *) LinearRecurrence[ {-6,-15,-20,-15,-6},{1,-6,21,-56,126},31] (* Harvey P. Dale, Oct 14 2016 *)

x='x+O('x^50); Vec(1/((1+x)^6-x^6)) \\ G. C. Greubel, Jul 02 2017

G.f.: 1/((1+x)^6-x^6).

a(n) = (-1)^n * Sum_{k=0..floor(n/6)} binomial(n+5,6*k+5). - Seiichi Manyama, Aug 05 2024

A085513 Number of "e"s in n (in English).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 3, 2, 2, 2, 2, 2, 4, 3, 3, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 0, 1, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 1, 1, 2, 3, 2, 4, 2, 3, 2, 4, 3, 3, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 2

Offset: 0

Samuel Thompson (samuelt(AT)ugcs.caltech.edu), Jul 03 2003

a(A006933(n)) = 0; a(A008520(n)) > 0. - Reinhard Zumkeller, Jan 23 2015

a(A006933(n)) = 0; a(A008520(n)) > 0; a(A121065(n)) = n and a(m) != n for m < A121065(n). - Reinhard Zumkeller, Jan 24 2015

			a(123) = 5 because "onE hundrEd twEnty-thrEE" has 5 e's.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Roel and Bas van Dijk, Numerals package, Hackage (Haskell packages)
Robert G. Wilson v, English names for the numbers from 0 to 11159 without spaces or hyphens .
Index entries for sequences related to number of letters in n

Cf. A005589.

Cf. A006933, A008520.

import Data.Maybe (fromJust)
import Data.Text (Text); import qualified Data.Text as T (unpack))
import Text.Numeral.Grammar.Reified (defaultInflection)
import qualified Text.Numeral.Language.EN as EN  -- see link
a085513 = length . filter (== 'e') . T.unpack . numeral where
   numeral :: Integer -> Text
   numeral = fromJust . EN.gb_cardinal defaultInflection
-- Reinhard Zumkeller, Jan 23 2015

StringCount[IntegerName/@Range[0,99],"e"] (* Ivan N. Ianakiev, Mar 25 2017 *)

from num2words import num2words
def A085513(n):
    return num2words(n).count('e') # Chai Wah Wu, Dec 20 2019

A000655 a(n) = number of letters in a(n-1), a(1) = 1 (in English).

Original entry on oeis.org

1, 3, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

N. J. A. Sloane

Decimal expansion of 1219/900. - Elmo R. Oliveira, May 05 2024

			One, three, five, four, four, four, ...

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A005589 (number of letters).

Cf. A061504 (French), A101432 (Spanish), A328263 (Polish).

Nest[Append[#, StringLength@ IntegerName[#[[-1]], "Words"]] &, {1}, 105] (* Michael De Vlieger, Feb 17 2021 *)

a(n) = 4 for n > 3, with a(1) = 1, a(2) = 3 and a(3) = 5. - Wesley Ivan Hurt, Oct 03 2020
E.g.f.: - 4 - 3*x - (1/2)*x^2 + (1/6)*x^3 + 4*exp(x). - Alejandro J. Becerra Jr., Feb 17 2021
G.f.: x*(1+2*x+2*x^2-x^3)/(1-x). - Elmo R. Oliveira, Jun 25 2024

A016037 Map numbers to number of letters in English name; sequence gives number of steps to converge (to 4).

Original entry on oeis.org

1, 3, 3, 2, 0, 1, 3, 2, 2, 1, 3, 4, 4, 3, 3, 3, 3, 2, 3, 3, 4, 2, 2, 5, 4, 4, 2, 5, 5, 4, 4, 2, 2, 5, 4, 4, 2, 5, 5, 4, 2, 3, 3, 4, 2, 2, 3, 4, 4, 2, 2, 3, 3, 4, 2, 2, 3, 4, 4, 2, 2, 3, 3, 4, 2, 2, 3, 4, 4, 2, 3, 4, 4, 5, 5, 5, 4, 5, 5, 5, 4, 2, 2, 5, 4, 4, 2, 5, 5, 4, 4, 2, 2, 5, 4, 4, 2, 5, 5, 4

Offset: 0

Robert G. Wilson v

The smallest n with a(n) = 7 is 1103323373373373373373373373373 (one nonillion one hundred and three octillion ...) which has 323 letters. - Roland Kneer, Jul 04 2013
From Robert G. Wilson v, Mar 13 2017: (Start)
First occurrence of k = 0,1,2,...: 4, 0, 3, 1, 11, 23, 323, 1103323373373373373373373373373, etc.;
0 only occurs at 4;
1 only occurs for n = 0, 5 & 9;
2 only occurs for n = 3, 7, 8, 17, 21, 22, 26, 31, 32, 36, 40, 44, 45, 49, 50, 54, 55, 59, 60, 64, 65, 69, 81, 82, 86, 91, 92 & 96;
3 occurs for n = 1, 2, 6, 10, 13, 14, 15, 16, 18, 19, 41, 42, 46, 51, 52, ..., ;
4 occurs for n =  11, 12, 20, 24, 25, 29, 30, 34, 35, 39, 43, 47, 48, 53, ..., ;
5 occurs for n =  23, 27, 28, 33, 37, 38, 73, 74, 75, 77, 78, 79, 83, 87, ..., ;
6 occurs for n = 323, 327, 328, 333, 337, 338, 374, 375, 379, 383, 387, ..., ; etc.
(End)
The basis of this sequence is that integers are named without the use of "and". As a minor correction at the time of this comment, therefore, the 31-digit number above beginning 1103 ... should be described as "one nonillion one hundred three octillion ...". If naming is done in accordance with UK English usage ("one hundred", "one hundred and one", ...), the first occurrence of a(n) = 0, 1, 2, 3, 4, 5, 6, 7, ... is for n = 4, 0, 3, 1, 11, 23, 124, 113373373373, ... Ian Duff, Dec 04 2019

			1 -> 3 -> 5 -> 4, so a(1) = 3.

(* get t from A005589 *) f[n_] := Length@ NestWhileList[ StringLength@ t[[# + 1]] &, n, UnsameQ, 2] - 2; Array[f, 100, 0] (* Robert G. Wilson v, Jun 01 2012 *)

Corrected at the suggestion of Kevin Ryde by Robert G. Wilson v, Jun 01 2012

A026858 Number of letters in n (in Italian).

Original entry on oeis.org

4, 3, 3, 3, 7, 6, 3, 5, 4, 4, 5, 6, 6, 7, 11, 8, 6, 11, 8, 10, 5, 7, 8, 8, 12, 11, 8, 10, 8, 9, 6, 8, 9, 9, 13, 12, 9, 11, 9, 10, 8, 10, 11, 11, 15, 14, 11, 13, 11, 12, 9, 11, 12, 12, 16, 15, 12, 14, 12, 13, 8, 10, 11, 11, 15, 14, 11, 13, 11, 12, 8, 10, 11, 11, 15, 14, 11, 13, 11, 12, 7, 9

Offset: 0

Alberto Maretti (alberto(AT)dis.uniroma1.it), Aug 03 2000

			Zero, uno, due, tre, quattro, cinque, sei, ...

Carmine Suriano, Table of n, a(n) for n = 0..10000
Index entries for sequences related to number of letters in n

Cf. A005589.

from num2words import num2words
def A026858(n): return len(num2words(n,lang='it'))  # Chai Wah Wu, Apr 28 2024

Corrected and extended by Marco Broglia, Oct 01 2008

Offset corrected to 0 ("zero") by Carmine Suriano, Jun 03 2011

cp's OEIS Frontend

A060403 Each term is the previous term plus the number of letters in the previous number, as conventionally spelled out in American English.

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A006944 Number of letters in the n-th ordinal number (in American English).

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A007005 Number of characters in the French spelling of n, including spaces and hyphens.

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A052362 Indices of records in length of English name of n including spaces and dashes (A052360): n such that k < n => A052360(k) < A052360(n).

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A073029 Names for numbers in American English, with each letter transformed into its index in the alphabet.

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A006090 Expansion of bracket function.

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A085513 Number of "e"s in n (in English).

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