A037196 -id:A037196 - cp's OEIS frontend

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

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

A037194 Number of vowels in n (in French).

Original entry on oeis.org

2, 1, 2, 2, 3, 1, 1, 1, 2, 2, 1, 2, 3, 3, 4, 3, 3, 2, 3, 3, 1, 3, 3, 3, 4, 2, 2, 2, 3, 3, 2, 4, 4, 4, 5, 3, 3, 3, 4, 4, 4, 6, 6, 6, 7, 5, 5, 5, 6, 6, 4, 6, 6, 6, 7, 5, 5, 5, 6, 6, 4, 6, 6, 6, 7, 5, 5, 5, 6, 6, 5, 7, 7, 7, 8, 7, 7, 6, 7, 7, 4, 5, 6, 6, 7, 5, 5

Offset: 0

Jean-Marc Rebert

			zéro 2, un 1, deux 2, trois 2, quatre 3, cinq 1, six 1, sept 1, huit 2, neuf 2, dix 1, onze 2, douze 3, treize 3, quatorze 4, quinze 3, seize 3, dix-sept 2, dix-huit 3, dix-neuf 3, vingt 1, vingt-et-un 3, vingt-deux 3, ...

Cf. A037196 (in American English).

More terms from Olivier Gérard, Jun 26 2001

a(60) onward corrected by Sean A. Irvine, Dec 16 2020

A037195 Number of consonants in the American English name of n.

Original entry on oeis.org

2, 1, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 4, 5, 4, 4, 4, 5, 4, 4, 4, 5, 6, 7, 6, 6, 6, 7, 7, 6, 4, 5, 6, 7, 6, 6, 6, 7, 7, 6, 3, 4, 5, 6, 5, 5, 5, 6, 6, 5, 3, 4, 5, 6, 5, 5, 5, 6, 6, 5, 3, 4, 5, 6, 5, 5, 5, 6, 6, 5, 4, 5, 6, 7, 6, 6, 6, 7, 7, 6, 3, 4, 5, 6, 5, 5, 5, 6, 6, 5, 3, 4, 5, 6, 5, 5, 5, 6, 6, 5, 6, 7, 8, 9

Offset: 0

N. J. A. Sloane

The letter "y" is being considered a vowel regardless of its usage in the word(s). [Corrected by Sean A. Irvine, Dec 16 2020]

Cf. A037196 (number of vowels).

More terms from David W. Wilson.

a(101)-a(103) corrected by Sean A. Irvine, Dec 16 2020

A332068 Numbers whose English name has at least two vowels and all the vowels are in alphabetical order.

Original entry on oeis.org

0, 3, 4, 7, 8, 11, 12, 17, 22, 23, 24, 26, 27, 28, 32, 34, 36, 42, 44, 52, 54, 56, 62, 64, 66, 70, 72, 73, 74, 76, 77, 78, 80, 82, 84, 86, 3000000, 3000002, 3000004, 3000040, 3000042, 3000044, 6000000, 6000002, 6000004, 6000040, 6000042, 6000044, 7000000, 7000002, 7000004

Offset: 1

M. F. Hasler, Aug 10 2020

Here (as in most OEIS sequences) vowel means one of the five letters A, E, I, O or U. (One could imagine variants that use Y, too.)
No number with "hUndrEd", "thoUsAnd", or "One / twO / foUr mIllioN" (or "fIvE, nInE"...) in it has the required property.
The vowels are counted with multiplicity: e.g., "thrEE" with two 'E's is listed.
The subsequence of numbers which have at least two distinct vowels in alphabetical order is 0, 4, 8, 22, 24, 26, 28, 32, 34, 44, 52, 54, 62, 64, 72, 74, 76, 78, 80, 82, 84, 86, 3000000, ...

			Numbers 0, 3, 4, ... have the required property, since their English names are "zErO", "thrEE", "fOUr", ...
Numbers 1, 2, 5, ... ("OnE", "twO", "fIvE", ...) don't have the property (vowels in incorrect order or less than two).

Cf. A052360.
A095947 \ {10} is the subset of numbers having only vowel E, and more than once.
Sequences related to vowels: A037196 (# vowels), A102869, A158352, A158354 (smallest number with n [distinct] vowels in AE / BE), A158353, A158355 (ditto, increasing), A058179 (all 5 vowels), A058180 (ditto, exactly once), A000852, A000861 (start/end with vowel), A019270, A080518 (self-describing), A059437, A079741, A152592, A174879, A241858.
See A332069 for numbers having all 5 vowels, in alphabetical order.

select( {is_A332068(n,v=Vec("aeiou"))=#(t=[c|c<-Vec(English(n)),setsearch(v,c)])>1&&t==vecsort(t)}, [0..999]) \\ See A052360 for English(). Insert "Set" after '#' to get the subset of numbers with > 1 distinct vowels.

A332069 Numbers whose American English name contains all 5 vowels in alphabetical order.

Original entry on oeis.org

1084, 1134, 1154, 1164, 1184, 1194, 1234, 1254, 1264, 1284, 1294, 1334, 1354, 1364, 1384, 1394, 1434, 1454, 1464, 1484, 1494, 1534, 1554, 1564, 1584, 1594, 1634, 1654, 1664, 1684, 1694, 1734, 1754, 1764, 1784, 1794, 1804, 1814, 1824, 1834, 1844, 1854, 1864, 1874, 1884, 1894

Offset: 1

M. F. Hasler, Aug 10 2020

The name of the number may contain other vowels (A, E, I, O or U) in any place and order. "American English" means that no "and" is used, e.g., 101 = "one hundred one".
Therefore (and because 1000 = "thousAnd" is the least number using the letter "A"), for any term a(n) < 10^4, the number a(n) + x*10^4 is also in the sequence for any x > 0, and so is any number a(n)*10^(6k) + m, m < 10^(6k), k > 0. (The statement isn't true with x*10^3: for example 1084 + 999000 does not have the letter "A".)
In French, 92 ("quAtrE-vIngt dOUze") has the property, and as a consequence the corresponding sequence consists mainly of 92 + x*100 with any x >= 0, and 472 + x*1000 with any x >= 0 ("quAtrE cent soIxante-dOUze"); there is no other term below 4000, from where on others (4012, 4061, 4062, ...) come into play.
In German, the first number to have an 'o' is "Million". Since the 'I' must be preceded by 'A' and 'E', the corresponding sequence would start only after 18*10^6: 18000005, 18000009, 18000015, 18000019, 18000021, 18000022, ...

			1084 is "(one) thousAnd EIghty fOUr". This is the smallest number whose English name contains all 5 vowels A, E, I, O, U in this order, therefore a(1) = 1084.

Anugata Sarkar, in reply to: What are some mind-blowing facts about mathematics?, quora.com, April / Oct. 2019.

Cf. A052360.
Sequences related to vowels: A037196 (# vowels), A102869, A158352, A158354 (smallest number with n [distinct] vowels in AE / BE), A158353, A158355 (ditto, increasing), A058179 (all 5 vowels), A058180 (ditto, exactly once), A000852, A000861 (start/end with vowel), A019270, A080518 (self-describing), A059437, A079741, A152592, A174879, A241858.
Cf. A332068 (also based on the order of vowels in the English name of numbers).

vowels=Vec("aeiou"); (isSubseq(a,b)=forvec(v=vector(#a,i,[i,#b]),vecextract(b,v)==a&&return(1),2)); select( {is_A332069(n)=#Set(n=[c|c<-Vec(English(n)),setsearch(vowels,c)])>4&&isSubseq(vowels,n)}, [0..2000]) \\ See A052360 for English().

A037200 Number of vowels in n (in German).

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 1, 3, 1, 2, 1, 1, 1, 3, 3, 2, 2, 3, 2, 3, 2, 5, 5, 5, 5, 4, 4, 6, 4, 5, 3, 6, 6, 6, 6, 5, 5, 7, 5, 6, 3, 6, 6, 6, 6, 5, 5, 7, 5, 6, 2, 5, 5, 5, 5, 4, 4, 6, 4, 5, 2, 5, 5, 5, 5, 4, 4, 6, 4, 5, 3, 6, 6, 6, 6, 5, 5, 7, 5, 6, 2, 5, 5, 5, 5, 4, 4, 6, 4, 5, 3, 6, 6, 6, 6, 5, 5, 7, 5, 6, 4

Offset: 0

N. J. A. Sloane

			null 1, eins 2, zwei 2, drei 2, vier 2, fünf 1, sechs 1, sieben 3, acht 1, neun 2, zehn 1, elf 1, zwölf 1, dreizehn 3, vierzehn 3, fünfzehn 2, sechszehn 2, siebzehn 3, achtzehn 2, neunzehn 3, zwanzig 2, ein-und-zwanzig 5, ...
a(100)=4 from "einhundert" rather than 2 from "hundert". - _Sean A. Irvine_, Dec 21 2020

Index entries for sequences related to German names of numbers

Cf. A037194 (in French), A037196 (in American English).

Cf. A037199 (number of consonants).

More terms from Olivier Gérard, Jun 26 2001
Missing a(19)=3 inserted by Sean A. Irvine, Dec 17 2020
a(100)=4 corrected by Sean A. Irvine, Dec 21 2020

A192295 Number of distinct consonants in the English name of n.

Original entry on oeis.org

2, 1, 2, 3, 2, 2, 2, 3, 3, 1, 2, 3, 4, 4, 4, 3, 4, 4, 4, 2, 4, 4, 4, 6, 6, 6, 6, 6, 6, 4, 4, 5, 5, 4, 5, 6, 6, 7, 5, 5, 4, 5, 5, 5, 4, 5, 6, 7, 6, 5, 3, 4, 4, 5, 4, 4, 5, 6, 5, 4, 4, 5, 5, 6, 6, 6, 4, 6, 6, 5, 5, 5, 6, 7, 7, 6, 6, 5, 7, 5, 4, 5, 5, 5, 6, 6, 6, 7, 4, 5, 3, 3, 4, 5, 5, 5, 5, 5, 5, 3, 4, 4, 6, 5, 5

Offset: 0

Kausthub Gudipati, Jun 27 2011

First differs from A037195 at the ninth term.
The letter "y" is here considered a consonant regardless of its usage in the word(s).
The maximum number of distinct vowels that can occur in the English name of n is 5, which occurs for n in A058179. - Jonathan Vos Post, Jul 12 2011

			a(13) = 4 because the distinct consonants present in THIRTEEN are T, H, R and N.
a(9) = 1 because N is the only consonant present in NINE.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A037195, A037196 (number of vowels in n), A058179 (numbers whose English names include all five vowels at least once).

Array[Length@ Union@ Select[Characters@ IntegerName@ #, And[LetterQ@ #, FreeQ[{97, 101, 105, 111, 117}, ToCharacterCode[#][[1]]]] &] &, 105, 0] (* Michael De Vlieger, Feb 15 2020 *)

More terms from Michael De Vlieger, Feb 15 2020

A321676 a(1)=1; for n > 1, a(n) is a(n-1) plus the number of vowels in the name of a(n-1) in US English.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 18, 22, 24, 27, 30, 31, 34, 37, 40, 41, 44, 47, 50, 51, 54, 57, 60, 61, 64, 67, 70, 72, 75, 79, 83, 87, 91, 95, 99, 103, 109, 115, 122, 128, 135, 142, 148, 155, 162, 168, 175, 183, 191, 199, 207, 212, 217, 224, 230, 234, 240, 244, 250

Offset: 1

Andrew Toothill, Dec 02 2018

This sequence takes into account the numbers written as words; for example, "fifty-seven" contains three vowels, so 3 is added to 57 to create the next term. The word "and" is not included in US English (cf. A158352), so 115 is written as "one hundred fifteen". This sequence is puzzling as it shares its first 6 terms with the odd numbers before jumping to 14, then 18. When only given the first 8 terms it can be very difficult to spot the rule.

Assumes "y" in e.g. "fifty" is not a vowel. - Chai Wah Wu, Dec 17 2018

Cf. A005589, A037196, A139282, A158352.

vowelCount[n_] := StringCount[IntegerName[n, "Words"], {"a","e","i","o","u"}]; f[n_] := n + vowelCount[n]; NestList[f, 1, 100] (* Amiram Eldar, Dec 10 2018 *)

from itertools import accumulate, repeat
from num2words import num2words
A321676_list = list(accumulate(repeat(1,100), lambda x, : x+sum(num2words(x).count(d) for d in 'aeiou'))) # _Chai Wah Wu, Dec 17 2018

a(n) = A139282(n) for n >= 2. - Chai Wah Wu, Dec 17 2018

More terms from Amiram Eldar, Dec 10 2018

cp's OEIS Frontend

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

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A037194 Number of vowels in n (in French).

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A037195 Number of consonants in the American English name of n.

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A332068 Numbers whose English name has at least two vowels and all the vowels are in alphabetical order.

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A332069 Numbers whose American English name contains all 5 vowels in alphabetical order.

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PARI

A037200 Number of vowels in n (in German).

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A192295 Number of distinct consonants in the English name of n.

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A321676 a(1)=1; for n > 1, a(n) is a(n-1) plus the number of vowels in the name of a(n-1) in US English.

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