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Statement

Definitions as on the Theorem 3 page: a positive integer $n$ is a $(2,2,2)$-number if $B(n^2)=2B(n)$, where $B$ is the binary digit sum, and $p_{(2,2,2)}(n)$ counts the $(2,2,2)$-numbers not exceeding $n$.

Theorem 4 (p. 259). The counting function of the $(2,2,2)$-numbers satisfies

$$p_{(2,2,2)}(n)\gg n^{0.0909}.$$

The paper gives no explicit constant. Its Conjecture 3 (p. 259), from the same independence heuristic as its Conjecture 2, proposes for each $k$ an asymptotic formula $p_{(2,k,k)}(n)=\frac{n}{(\log n)^{1/2}}G_k+R(n)$ with $G_k=\sqrt{2\log2/(\pi(k^2+k))}$ and $R(n)=o(n/(\log n)^{1/2})$.

Source. Theorem 4 and Conjecture 3, p. 259, 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 statement was read on p. 259. The paper gives no proof beyond saying that it follows by a procedure analogous to the one for Theorem 3 (p. 258); the proof is not checked here.

Proof pointer

The paper says only that an analogous procedure to the outline for Theorem 3 proves it. For the details of the Theorem 3 construction it refers to G. Melfi, On simultaneous binary expansion of $n$ and $n^2$, arXiv:math/0402458; it names no separate source for Theorem 4.

Dependencies

The method of Theorem 3.

Bears on

No Erdős problem in this corpus.