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It will be proven that 2023 is the only real quantity that equals the sum of its digits multiplied twice by the sum of the squares of its digits. Indeed:
2023 = (2+0+2+3) x (2^2 + 0^2 + 2^2 + 3^2)^2
Are most different numbers n happier (the corresponding product is bigger than n) or sadder (the corresponding product is lower than n) than n?
requested 49 minutes in the past
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All numbers larger than some threshold are
sadder than the quantity. Intuitively the sum of digits is roughly the logarithm of the quantity and the sum of squares of the digits squared is an influence of the logarithm of the quantity, so the calculation is roughly an influence of the logarithm of the quantity, which grows extra slowly than the quantity itself.
To make a certain on the edge
Let $n$ have $ok$ digits. The proper facet is at most $9kcdot (81k)^2=3^6k^3$ The left facet is at the least $10^{k-1}$. For $ok=7$ the left is $1,000,000$ and the suitable is $250,047$ so all numbers of at the least $7$ digits are sadder than the quantity.
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