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A158352 a(n) is the smallest number whose name in US English contains n vowels.

%I A158352 #36 Dec 04 2020 19:39:00
%S A158352 2,1,11,14,71,101,111,114,171,1071,1101,1111,1114,1171,11171,14171,
%T A158352 71171,101171,111171,114171,171171,1071171,1101171,1111171,1114171,
%U A158352 1171171,11171171,14171171,71171171,101171171,111171171,114171171,171171171,1071171171,1101171171
%N A158352 a(n) is the smallest number whose name in US English contains n vowels.
%C A158352 In US English, "101" is written as "one hundred one".
%C A158352 From _Michael S. Branicky_, Oct 24 2020 (Start)
%C A158352 The sequence counts vowels by counting the instances of the letters 'a', 'e', 'i', 'o', 'u', and 'y' (see Example). If 'y' were not included, then a(5) = 102 ("OnE hUndrEd twO") is the first among subsequent value changes.
%C A158352 In extending the sequence to large numbers, the "American system" (Weisstein link), also known as the "short scale" (Wikipedia link), was used. Also, the common written form is adopted ("one thousand one hundred seventeen" not "eleven hundred seventeen"; Wilson link).
%C A158352 Also, a(n) = A158353(n) for n >= 3. Proof. For them to differ after (171)^n, which has 9*n vowels, would require either (i) "one [power name] two" be next, preceding "one [power name] one"), which could only occur if [power name] had 9*n-2 vowels; or (ii) [power name] has > 9*n-2 vowels and a subsequent power name has less (within a range depending on n). Neither case occurs for existing names (see Wikipedia Large numbers link). (End)
%H A158352 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LargeNumber.html">Large Number</a>
%H A158352 Wikipedia, <a href="https://en.wikipedia.org/wiki/Names_of_large_numbers">Names of Large Numbers</a>
%H A158352 Wiktionary, <a href="https://en.wiktionary.org/wiki/one_hundred_one">one hundred one</a> (US)
%H A158352 Wiktionary, <a href="https://en.wiktionary.org/wiki/one_hundred_and_one">one hundred and one</a> (UK)
%H A158352 Robert G. Wilson v, <a href="https://oeis.org/A001477/a001477.txt">American English names for the numbers from 0 to 100999 without spaces or hyphens</a>
%e A158352 "twO" has 1 vowel, "OnE" has 2, "ElEvEn" has 3, "fOUrtEEn" has 4, "sEvEntY-OnE" has 5, "OnE hUndrEd OnE" has 6, "OnE hUndrEd ElEvEn" has 7.
%t A158352 With[{s = Array[StringCount[StringJoin@ Map[If[IntegerQ[#], IntegerName[#], ToString[#]] &, ToExpression@ StringSplit@ IntegerName[#]], {"a", "e", "i", "o", "u", "y"}] &, 10^4]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* _Michael De Vlieger_, Nov 06 2020 *)
%o A158352 (Python)
%o A158352 from num2words import num2words
%o A158352 def A158352(n):
%o A158352   i = 1
%o A158352   while sum(1 for c in num2words(i).replace(' and ', '') if c in "aeiouy")!=n:
%o A158352     i += 1
%o A158352   return i
%o A158352 print([A158352(n) for n in range(1,16)]) # _Michael S. Branicky_, Oct 23 2020
%Y A158352 Cf. A000052, A079741, A158353.
%K A158352 nonn,word
%O A158352 1,1
%A A158352 _Rodolfo Kurchan_, Mar 16 2009
%E A158352 Edited by _Jon E. Schoenfield_, Sep 29 2018
%E A158352 a(9) onwards from _Michael S. Branicky_, Oct 23 2020