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Statement

Definition 1 (p. 258). For integers $k\ge2$, $l\ge1$ and $m\ge2$, a positive integer $n$ is a $(k,l,m)$-number if the sum of the base-$k$ digits of $n^m$ is $l$ times the sum of the base-$k$ digits of $n$. The counting function $p_{(k,l,m)}(n)$ is the number of $(k,l,m)$-numbers not exceeding $n$ (p. 258).

With $B(n)$ the binary digit sum of $n$, the $(2,1,2)$-numbers are the $n$ with $B(n)=B(n^2)$ and the $(2,2,2)$-numbers those with $2B(n)=B(n^2)$.

Theorem 3 (p. 258). The counting function of the $(2,1,2)$-numbers satisfies

$$p_{(2,1,2)}(n)\gg n^{0.025}.$$

The paper gives no explicit constant. On p. 259 it reports, as announced by Sándor in a personal communication (2003), the upper bound $p_{(2,1,2)}(n)\ll n^{0.9183}$, and it states as Conjecture 2, from a heuristic treating $B(n)$ and $B(n^2)$ as independent, that $p_{(2,1,2)}(n)=n^{\alpha+o(1)}$ with $\alpha=\log1.6875/\log2\approx0.7548875$.

Source. Definition 1 and Theorem 3, p. 258, of Giuseppe Melfi, On certain positive integer sequences, Riv. Mat. Univ. Parma (7) 3* (2004), 253--260, as identified on the source card.

Read depth. Claims checked: the definition, the statement and the remarks on p. 259 were read clause by clause. The paper gives only an outline of the proof and refers to G. Melfi, On simultaneous binary expansion of $n$ and $n^2$, arXiv:math/0402458, for details; that proof is not checked here.

Proof pointer

Page 258, outline only. For every $n$ one builds $n$ distinct $(2,1,2)$-numbers not exceeding $An^{40}$, for a constant $A$; this gives the exponent $1/40=0.025$. The construction starts from an arbitrary number not exceeding $n$ and adds a suitable finite string of zeros and ones to its binary expansion, controlling $B$ of the new number and of its square at once; it uses the identity $B(n(2^\nu-1))=\nu$ for $n<2^\nu$.

Dependencies

The full proof is in the arXiv preprint math/0402458 cited above.

Bears on

No Erdős problem in this corpus.