You might not be happy that someone investigated this because you apparently made up this example in order to make fun of this kind of question, but I looked into this a little bit and it's more subtle than I first thought.
The typical random integer small enough to have a well-defined English name (following Guy and Conway's prefixes at https://en.wikipedia.org/wiki/Names_of_large_numbers#Extensi... which will go up to 10³⁰⁰⁰-1) has a significantly longer English name than its decimal expansion. E.g. if we take fifty random digits like
96689907044105050355231217362382284051155144740067
the name of this number is "ninetysixquindecillionsixhundredeightyninequattuordecillionninehundredseventredecillionfortyfourduodecilliononehundredfiveundecillionfiftydecillionthreehundredfiftyfivenonilliontwohundredthirtyoneoctilliontwohundredseventeenseptillionthreehundredsixtytwosextillionthreehundredeightytwoquintilliontwohundredeightyfourquadrillionfiftyonetrilliononehundredfiftyfivebilliononehundredfortyfourmillionsevenhundredfortythousandsixtyseven"
or 430 letters, while the number is written with only 50 digits.
For small numbers, the closest approach between the lengths of the two is 10 (len("ten") == 3, len("10") == 2). And the typical word length grows faster than the digit length.
But the lengths do cross over many times up in the illions, as the illion terminology is unusually short compared to the size of the number described. Specifically, the smallest integer where this occurs is 1,000,000,000, where len("onebillion") == len("1000000000"). That's the smallest member of your set.
There are other examples such as 10¹³+1, 10¹³+2, 10¹³+6:
len("tentrillionone") == len("tentrilliontwo") == len("tentrillionsix") == len("10000000000001")
And there are many more further out into the illions.
For any example in which "one", "two", or "six" appears in the name of a number, any of the others can be substituted and the property will still hold. Likewise for "four"-"five"-"nine" and "three"-"seven"-"eight".