On simultaneous binary expansions of n and n²
classification
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binarydigitsfamilysequenceaccuratelyasymptoticcomposedconjecture
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A new family of sequences is proposed. An example of sequence of this family is more accurately studied. This sequence is composed by the integers $n$ for which the sum of binary digits is equal to the sum of binary digits of $n^2$. Some structure and asymptotic properties are proved and a conjecture about its counting function is discussed.
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