11 problems

A sharpening of our old conjecture with Turán would state: If $a_n < Cn^2$ for all $n$ then $\limsup f(n) = \infty$. In fact, for what functions $g(n) \to \infty$ does…

The sequence B is called a basis of order $k$, if every integer is the sum of $k$ or fewer $b$-s. Let $d_s(\mathrm{A}) = \alpha$ then I proved that … Thus every base is an essentia…

Let $b_1<b_2<\ldots$ be an infinite sequence of integers such that every integer is of the form $b+k^2$ for some member $b$ of the sequence. Let $B(x)$ denote the number of members…

Let $a_1<a_2<\cdots$ be an infinite sequence of integers. Straus and I conjectured that there is a sequence of density $0$, $b_1<b_2<\cdots$, such that every integer is of the form…

Let $a_1 < a_2 < \ldots$ be any infinite sequence of integers. Straus and I conjectured that there always exists a sequence $b_1 < b_2 < \ldots$ of density $0$, so that every suffi…

Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence of integers, and denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Also I offer 100 dollars for an explicit cons…

Let $A = \{a_1 < a_2 < \ldots\}$ be an infinite sequence of integers, and denote by $f(n)$ the number of solutions of $n = a_i + a_j$. Also Turán and I conjectured that if…

Perhaps the following rather silly conjecture could be added. Is it true that the set of odd integers not of the form $2^k + p$ is the not necessarily disjoint union of an infinite…

Are there infinitely many odd integers $n$ for which $n - 2^u$, $1 \le 2^u \le n$ is never squarefree? In fact is there any such an integer?

One could ask the following (probably unattackable) problem. Is it true that there is an $r$ so that every integer is the sum of a prime and $r$ or fewer powers of 2.

Crocker [16] proved that there are infinitely many odd integers not of the form $2^{k_1} + 2^{k_2} + p$, but his proof only gives that the number of integers $\leq x$ not of the fo…