A085513 -id:A085513 - cp's OEIS frontend

A006933 'Eban' numbers (the letter 'e' is banned!).

2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, 2006, 2030, 2032, 2034, 2036, 2040, 2042, 2044, 2046, 2050, 2052, 2054, 2056, 2060, 2062, 2064, 2066, 4000, 4002, 4004, 4006, 4030, 4032, 4034, 4036, 4040, 4042, 4044, 4046, 4050, 4052, 4054, 4056, 4060, 4062, 4064, 4066, 6000

Offset: 1

N. J. A. Sloane

Invented by N. J. A. Sloane circa 1990.
Theorem (N. J. A. Sloane): in English every odd number contains an 'e'.
The first number that would appear in the British Eban list but not the American list is 2*10^21. - Douglas Boffey, Jun 21 2012
A085513(a(n)) = 0. - Reinhard Zumkeller, Jan 23 2015

			2052 is in the sequence because written out in English words, "two thousand fifty-two", it does not contain a single instance of the letter E.
2053 (two thousand fifty-three) is not in the sequence because written out it contains two instances of E.

J. C. Hernandez et al., "Characterization of Eban numbers", pp. 197-200, Journal of Recreational Mathematics, 31 (3) 2002-2003.
Georges Perec, La disparition, Editions Gallimard, Paris, 1969; English translation: A Void, Harvill, 1994. (A novel that does not use the letter "e".)
Georges Perec, Les Revenentes [a novel in which the only vowel that appears is 'e']. - From Simon Plouffe, Mar 12 2010
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Roel and Bas van Dijk, Numerals package, Hackage (Haskell packages)
Brady Haran and N. J. A. Sloane, What Number Comes Next? (2018), Numberphile video
Eric Weisstein's World of Mathematics, Eban Number
Index entries for sequences related to number of letters in n

Cf. A082504.
Cf. A085513, A008520 (complement), A008521 (ban o), A008523 (ban t), A089589 (ban i), A089590 (ban u), A014254 (a French version), A287876 (a Hebrew version).
Cf. A008537 (without 'n'), A072956 (turban numbers: without r, t or u), A072957 (urban numbers: without r or u), A089589 (without 'i').

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
a006933 n = a006933_list !! (n-1)
a006933_list = filter (T.all (/= 'e') . numeral) [0..] where
   numeral :: Integer -> Text
   numeral = fromJust . EN.gb_cardinal defaultInflection
-- Reinhard Zumkeller, Jan 23 2015

[ n : n in [1..100] | forall{ i : i in [1..#seq] | seq[i] in eban[(i-1)mod 3+1]} where seq is Intseq(n) ] where eban is [[0,2,4,6],[0,3,4,5,6],[0]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

is(n)=!setsearch(Set(Vec(English(n))), "e") \\ See A052360 for English(). - M. F. Hasler, Apr 01 2019

from num2words import num2words
[n for n in range(6001) if 'e' not in num2words(n)] # Indranil Ghosh, Jul 05 2017

More terms from WG Zeist, Aug 28 2012

More cross-references from M. F. Hasler, Apr 01 2019

A008520 Numbers whose American English name contains the letter 'e'.

Original entry on oeis.org

0, 1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37, 38, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 63, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

N. J. A. Sloane

A085513(a(n)) > 0. - Reinhard Zumkeller, Jan 23 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Roel and Bas van Dijk, Numerals package, Hackage (Haskell packages)
Index entries for sequences related to number of letters in n

Cf. A006933 (complement), A085513.

Cf. A008519 (o), A008522 (t), A008536 (n), A008538 (s), A008540 (f), A008553 (y).

import Data.Maybe (fromJust)
import Data.Text (Text); import qualified Data.Text as T (any)
import Text.Numeral.Grammar.Reified (defaultInflection)
import qualified Text.Numeral.Language.EN as EN  -- see link
a008520 n = a008520_list !! (n-1)
a008520_list = filter (T.any (== 'e') . numeral) [0..] where
   numeral :: Integer -> Text
   numeral = fromJust . EN.gb_cardinal defaultInflection
-- Reinhard Zumkeller, Jan 23 2015

A008520Q[n_]:=StringContainsQ[IntegerName[n,"Words"],"e"];Select[Range[0,200],A008520Q] (* Paolo Xausa, Aug 11 2023 *)

Name edited by Michael De Vlieger, Aug 11 2023

A036448 Smallest positive number containing n e's when spelled out in US English.

Original entry on oeis.org

2, 1, 3, 11, 17, 111, 117, 317, 1317, 3317, 11317, 17317, 111317, 117317, 317317, 1317317, 3317317, 11317317, 17317317, 111317317, 117317317, 317317317, 1317317317, 3317317317, 11317317317, 17317317317, 111317317317, 117317317317, 317317317317, 1317317317317, 3317317317317

Offset: 0

Rodolfo Kurchan

From Michael S. Branicky, Oct 24 2020: (Start)
"US English" connotes that no "and" is used ("one hundred one") and, importantly here, that the names of large numbers follow the "American system" (Weisstein link), also known as the short scale (Wikipedia link). The previous a(8) and a(9) were based on "eleven hundred and seventeen" and "seventeen hundred and seventeen", which are less common written forms (Wikipedia English numbers link). To make the sequence precise, the common written form is adopted ("one thousand one hundred seventeen"; Wilson link; A000052 Example). Thus, a(n) is the least m such that A085513(m)=n.
The sequence follows the pattern of 1(317)^n, 3(317)^n, 11(317)^n, 17(317)^n, 111(317)^n, 117(317)^n, 317(317)^n for n = 0 through 7 and whenever the largest named power has no "e". a(50) > 10^21 = "one sextillion" which is the first power name that has an "e", breaking the pattern. In that case, a(50) = 1117(317)^6 and a(51) = 1(317)^7. Whenever the largest power has 1 "e" it follows this pattern. If it has m>1 "e"'s, the first block of three is shifted lower to a(7-m). See Wikipedia link for Names of large numbers for power names.
(End)

			One has 1 e.
Three has 2 e's.

Rodolfo Marcelo Kurchan, Problem 1882, Another Number Sequence, Journal of Recreational Mathematics, vol. 23, number 2, p. 141.

Eric Weisstein's World of Mathematics, Large Number
Wikipedia, English numerals
Wikipedia, Names of Large Numbers
Robert G. Wilson v, American English names for the numbers from 0 to 100999 without spaces or hyphens

Cf. A000027, A000052, A001477, A005589, A006933, A037196, A085513.

from num2words import num2words
def A036448(n):
    i = 1
    while num2words(i).count("e")!=n:
        i += 1
    return i
print([A036448(n) for n in range(1,12)]) # Michael S. Branicky, Oct 23 2020

a(8)-a(9) changed and a(11)-a(30) added by Michael S. Branicky, Oct 23 2020

a(0)=2 inserted by Sean A. Irvine, Nov 02 2020

A121065 a(n) is the smallest number in English which contains n letter 'E's.

Original entry on oeis.org

2, 0, 3, 11, 17, 111, 117, 317, 1317, 3317, 11317, 17317, 111317, 117317, 317317, 1317317, 3317317, 11317317, 17317317, 111317317, 117317317, 317317317, 1317317317, 3317317317, 11317317317, 17317317317, 111317317317, 117317317317, 317317317317, 1317317317317

Offset: 0

Ray G. Opao, Aug 10 2006

4, 5, 6, 8, 9 never appear in any of these numbers because in each case there is a smaller digit with the same number of e's. 2 (the smallest number with no e's) never appears in any term after a(0). - Sean A. Irvine, Nov 10 2009

A085513(a(n)) = n and A085513(m) != n for m < a(n). - Reinhard Zumkeller, Jan 24 2015

			a(2) = THREE, which has two Es.

Index entries for sequences related to number of letters in n

Cf. A085513, A008520, A006933. Variant of A036448.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a121065 = fromJust . (`elemIndex` a085513_list)
-- Reinhard Zumkeller, Jan 24 2015

From Chai Wah Wu, Dec 20 2019: (Start)
a(n) = a(n-1) + 1000*a(n-7) - 1000*a(n-8) for n > 9 (conjectured).
G.f.: (-1000*x^9 + 3000*x^8 - 1800*x^7 + 6*x^6 + 94*x^5 + 6*x^4 + 8*x^3 + 3*x^2 - 2*x + 2)/((x - 1)*(1000*x^7 - 1)) (conjectured). (End)

More terms Sean A. Irvine, Nov 10 2009
a(19) - a(21) added by Reinhard Zumkeller, Jan 24 2015
a(22) - a(29) from Chai Wah Wu, Dec 20 2019

A191784 Number of e's in the English name of the n-th odd number.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 2, 4, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 5, 4, 4, 6, 5, 4, 5, 4, 5, 4, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 3, 5, 6, 5, 6, 5, 4, 5, 4, 5, 4, 4, 5, 4, 5, 4, 2, 3, 2, 3, 2

Offset: 1

Kausthub Gudipati, Jun 25 2011

Every odd number has the letter e in its English name, so a(n) can never be 0.

			a(5) = 1, because the 5th odd number is "nine", which contains one "e".

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Cf. A085513.

units:=[1,0,2,0,1,0,2,1,1,1,3,2,2,2,2,2,4,3,3]:tens:=[0,0,1,0,0,0,0,2,1,1]: A191784 := proc(n) global tens,units: if(n<=10)then return units[2*n-1]: elif(n<=50)then return units[2*((n-1) mod 5) + 1] + tens[floor((n-1)/5)+1]: elif(n<=500)then return 1+units[floor((n-1)/50)]+procname(((n-1) mod 50) + 1): fi: return units[floor((n-1)/500)]+procname(((n-1) mod 500) + 1): end: seq(A191784(n),n=1..105); # valid up to n=5000, Nathaniel Johnston, Jun 26 2011

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A006933 'Eban' numbers (the letter 'e' is banned!).

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A008520 Numbers whose American English name contains the letter 'e'.

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A036448 Smallest positive number containing n e's when spelled out in US English.

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A121065 a(n) is the smallest number in English which contains n letter 'E's.

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A191784 Number of e's in the English name of the n-th odd number.

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