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

Author

Rodolfo Kurchan

Keywords

• nonn
• word

Comments

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)

Examples

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

References

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

Links

• 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

Crossrefs

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

Programs

Python

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

Extensions

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