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In the 2 minute BBC News audio clip Sesame Street: What is Count von Count's favourite number? "The Count" is asked

Do you have a favorite number?

to which he replied

Thirty four thousand, nine hundred and sixty nine. It's a square root thing.

Does the number 34,969 have any particularly notable properties that would make it "The Count"'s favorite number?

Revelations from the world of counting from the late Jerry Nelson, the voice of Count von Count, who was interviewed by Tim Harford to mark Sesame Street's 40th anniversary in 2009.

Photo: Count von Count attends Macy's Thanksgiving Parade 2018 Credit: Getty Images

Release date: 01 February 2019
Duration: 2 minutes

There are some speculations in the following article:

https://www.bbc.com/news/magazine-19409960

The following is taken verbatim from the link:

34,969 is 187 squared. But why 187?

More or Less turned to its listeners for help.

Toby Lewis noted that 187 is the total number of points on the tiles
of a Scrabble game, speculating that the Count might have counted
them.

David Lees noticed that 187 is the product of two primes - 11 and 17 -
which makes 34,969 a very fine number indeed, being 11 squared times
17 squared. What, he asked, could be lovelier?

And Simon Philips calculated that 187 is 94 squared minus 93 squared -
and of course 187 is also 94 plus 93 (although that would be true of
any two consecutive numbers, as reader Lynn Wragg pointed out). An
embarrassment of riches!

But both he and Toby Lewis hinted at darkness behind the Count's
carefree laughter and charming flashes of lightning: 187 is also the
American police code for murder.

Murder squared: was the Count trying to tell us something?

Curiously,
$$sqrt{1234567890}=35136.418ldots$$
which is kinda-sorta close to $34969$, although it doesn't seem close enough to make the joke work.

Perhaps it's that $34969=187^2$ is the largest perfect square whose own square doesn't exceed $1234567890$ (since $sqrt[4]{1234567890} = 187.447ldots$).

Considering how the Count counts, one might think his favorite number relates to
$$12345678910$$
It's perhaps worth noting that
$$begin{align}
sqrt{12345678910} &;=; 111,111.11ldots \
sqrt[4]{12345678910} &;=; phantom{111,}333.33333ldots
end{align}$$
where I have conveniently truncated the digits for best effect.

Unrelatedly: I've always been a little disappointed that the Count's full name is "Count von Count" instead of, say, "Count von Tuthrifore".

Presumably if it's a square root thing, we're allowed to consider its square root, which, as other answers have noted, is $187$.

According to David Wells, The Penguin Dictionary of Curious and Interesting Numbers, (Penguin, 1986), $187$ is

The smallest of a group of $3$-digit numbers that require $23$ reversals to form a palindrome.

This is then followed by the entry for $196$, which includes an explanation of palindromes by reversal. The procedure is to reverse a number's digits and add the resulting number to the original.

Do all numbers become palindromes eventually? The answer to this problem is not known. $196$ is the only number less than $10,000$ that by this process has not yet produced a palindrome. P. C. Leyland has performed $50,000$ reversals, producing a number of more than $26,000$ digits with no palindrome appearing, and P. Anderton has taken this up to $70,928$ digits, also without success.

Most of the $3$-digit numbers require only a few reversals. The ones needing $23$ are

$${187, 286, 385, 583, 682, 781, 880},$$

all of which go via $968$ then $1837$, and end up at $8813200023188$.

So my guess is that the Count is able to take the square root of $34969$ in his head then do the $23$ reversals and get the palindrome, but hasn't yet managed to do the same with $38416=196^2$.