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It may 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. Certainly:
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 larger than n) or sadder (the corresponding product is lower than n) than n?
requested 1 hour in the past
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All numbers better 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 sure on the brink
Let $n$ have $ok$ digits. The suitable facet is at most $9kcdot (81k)^2=3^6k^3$ The left facet is no less than $10^{k-1}$. For $ok=7$ the left is $1,000,000$ and the correct is $250,047$ so all numbers of no less than $7$ digits are sadder than the quantity.
answered 19 minutes in the past
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