Let $f$ be a positive multiplicative function and let $k\geq 2$ an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in sense $\sum_{p}|f(p)-1|=\infty$, there exists real number $c>0$ such $k$-tuples $(f(p+1),\ldots,f(p+k))$ are dense hypercube $[0,c]^k$ or $[c,+\infty)^k$. In particular, $f(p+1),\ldots,f(p+k)$ can put any increasin...

The concept of antimagic labelings a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture that every connected graph, except single edge, antimagic. Some graphs are known be antimagic, but little has been about sparse graphs, not even trees. This paper studies weak version called k-shifted-antimagic which allow the ...