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### Number of 1 in binary representation of n

Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.
For example, $13=1101_2$ so $1(13)=3\\$
Is there explicit form of $\,\,\sum{1(i)x^i} $?
I checked OEIS and didn't find ...

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### Distribution of digits of $pq$-adic idempotents (aka “automorphic numbers”)

Let $p$ and $q$ be distinct primes. By the ring of $pq$-adic integers I mean the ring $\mathbb{Z}_{pq} := \varprojlim \mathbb{Z}/(pq)^n\mathbb{Z}$ which is obviously isomorphic to $\mathbb{Z}_p ...

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### Square and reversed integer

For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$,
we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= ...

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### sum of digits in different bases

Given a natural number, What is the maximal natural number below it, whose sums of digits in base 10 and base 2 are the same? Is there a clever algorithm to do this aside from the brute force search? ...

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### Here is a generalization of n-ary base notation for numbers. Surely unoriginal. Anybody know where to find literature on it?

If $f:\mathbb{N}\to\mathbb{N}$ is any strictly increasing function with $f(0)=1$, define the base $f$ notation for natural numbers inductively as follows:
$0$ is represented as $()$ (the empty ...

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### Lower bound on # of nonzero digits in ternary expansions of powers of 2?

Does anyone know of any lower bounds on the number of nonzero digits that appear in powers of 2 when written to base 3? (Other than the easy "If it's more than 8 it has to have at least 3.") I know ...

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### Can N^2 have only digits 0 and 1, other than N=10^k?

Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer N, not of the form 10^k, such that the digits of N^2 are all 0s and 1s?
It seems very unlikely, but I ...