A059437 -id:A059437 - cp's OEIS frontend

A000861 Numbers ending with a vowel in American English.

0, 1, 2, 3, 5, 9, 12, 21, 22, 23, 25, 29, 31, 32, 33, 35, 39, 41, 42, 43, 45, 49, 51, 52, 53, 55, 59, 61, 62, 63, 65, 69, 71, 72, 73, 75, 79, 81, 82, 83, 85, 89, 91, 92, 93, 95, 99, 101, 102, 103, 105, 109, 112, 121, 122, 123, 125, 129, 131, 132, 133, 135, 139, 141, 142

Offset: 1

N. J. A. Sloane

The German name of n ends in a vowel iff n==2 or n==3 (mod 100), so the German analog of this sequence would be {2, 3, 102, 103, 202, 203, ...}. - M. F. Hasler, Sep 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A247359 (complement), A152592 (characteristic fct of the complement), A059437 (a variant of the latter: vowels or "y").

Cf. A247361 (French analog) and A247360 (complement).

print1(0",");for(n=1,3,for(k=10^(n-1),10^n-1,m100=k%100;m10=k%10;if(m100==12 || ((m100>20 || m100<10) && ((m10==1)||(m10==2)||(m10==3)||(m10==5)||(m10==9))),print1(k",")))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006

is(n,S=[1,2,3,5,9,12])=setsearch(S,if(n%100>19,n%10,n%100))||!n \\ Add 0 in S (and optionally remove trailing ||...) for the A059437 variant. Add "Set()" around [...] for PARI versions < 2.6. - M. F. Hasler, Sep 14 2014

From Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) + a(n-46) - a(n-47) for n > 48.
G.f.: x^2*(x^46 + 4*x^45 + 2*x^44 + x^43 + x^42 + 2*x^41 + 4*x^40 + 2*x^39 + x^38 + x^37 + 2*x^36 + 4*x^35 + 2*x^34 + x^33 + x^32 + 2*x^31 + 4*x^30 + 2*x^29 + x^28 + x^27 + 2*x^26 + 4*x^25 + 2*x^24 + x^23 + x^22 + 2*x^21 + 4*x^20 + 2*x^19 + x^18 + x^17 + 2*x^16 + 4*x^15 + 2*x^14 + x^13 + x^12 + 2*x^11 + 4*x^10 + 2*x^9 + x^8 + x^7 + 9*x^6 + 3*x^5 + 4*x^4 + 2*x^3 + x^2 + x + 1)/(x^47 - x^46 - x + 1). (End)

Corrected by Logan K. Young (lkyfella(AT)yahoo.com), Mar 19 2005

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006

A037196 Number of vowels in the American English name of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

"American English" means that there is no "and" in the names of numbers, cf. example. - M. F. Hasler, Aug 26 2020

			a(20) = 1 for "twEnty" with 1 vowel: 'y' does not count.
a(101) = 6 for "OnE hUndrEd OnE" with 6 vowels: no "and" as in the "British" variant "one hundred and one" which would have 7 vowels.

Cf. A005589, A052360 (number of letters in English name of numbers with/without spaces and dashes).

Sequences related to 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, A332068, A332069.

vowels=Vec("aeiou"); apply( {A037196(n)=#[c|c<-Vec(English(n)),setsearch(vowels,c)]}, [0..104]) \\ see A052360 for English(). - M. F. Hasler, Aug 26 2020

from num2words import num2words
def a(n): return sum(1 for c in num2words(n).replace(" and", "") if c in "aeiou")
print([a(n) for n in range(105)]) # Michael S. Branicky, Mar 23 2025

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000
Name edited and crossrefs added by M. F. Hasler, Aug 26 2020
a(19)=4 corrected by Sean A. Irvine, Dec 16 2020

A152592 Consider the last letter of each of the English words zero, one, two, three, four, five, ... . Write down 0 for a vowel {a,e,i,o,u}, 1 for a consonant.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1

Offset: 0

Paul Curtz, Dec 09 2008

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,-1,1).

See A059437 for another version.

From Chai Wah Wu, Apr 18 2024: (Start)
a(n) = a(n-1) - a(n-5) + a(n-6) for n > 25.
G.f.: x^4*(x^21 - x^20 + x^17 - x^14 + x^13 - x^11 + x^10 - x^9 + x^8 - x^7 - x^2 + x - 1)/(x^6 - x^5 + x - 1). (End)

Edited by N. J. A. Sloane, Apr 09 2009, Apr 11 2009

Extended by Nathaniel Johnston, May 05 2011

A247359 Numbers whose English name ends in a consonant.

Original entry on oeis.org

4, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 24, 26, 27, 28, 30, 34, 36, 37, 38, 40, 44, 46, 47, 48, 50, 54, 56, 57, 58, 60, 64, 66, 67, 68, 70, 74, 76, 77, 78, 80, 84, 86, 87, 88, 90, 94, 96, 97, 98, 100, 104, 106, 107, 108, 110, 111, 113, 114, 115, 116, 117

Offset: 1

M. F. Hasler, Sep 14 2014

Complement of A000861.

Sequence A152592 is the characteristic function of this set, A059437 would yield a variant including multiples of 10: 20, 30, ..., 90, 120, 130, ...

Cf. A247359.

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().

A247362 Interleave A000861 and A247359, numbers ending in a vowel resp. consonant in English.

Original entry on oeis.org

0, 4, 1, 6, 2, 7, 3, 8, 5, 10, 9, 11, 12, 13, 21, 14, 22, 15, 23, 16, 25, 17, 29, 18, 31, 19, 32, 20, 33, 24, 35, 26, 39, 27, 41, 28, 42, 30, 43, 34, 45, 36, 49, 37, 51, 38, 52, 40, 53, 44, 55, 46, 59, 47, 61, 48, 62, 50, 63, 54, 65, 56, 69, 57, 71, 58, 72

Offset: 0

M. F. Hasler, Sep 14 2014

A permutation of the nonnegative integers.

See A247361 for the analog permutation of the positive integers, starting with a(1)=1.

Cf. A152592, A059437.

cp's OEIS Frontend

A000861 Numbers ending with a vowel in American English.

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A037196 Number of vowels in the American English name of n.

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A152592 Consider the last letter of each of the English words zero, one, two, three, four, five, ... . Write down 0 for a vowel {a,e,i,o,u}, 1 for a consonant.

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A247359 Numbers whose English name ends in a consonant.

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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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A247362 Interleave A000861 and A247359, numbers ending in a vowel resp. consonant in English.

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