𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On Gaps Between k-Free Numbers

✍ Scribed by O. Trifonov

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
337 KB
Volume
55
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

An elementary proof is given that for (k \geqslant 3) there exists a constant (c=c(k)) such that for (x) sufficiently large (depending on (k) ), the interval (\left(x, x+c x^{1,12 k+1} \log x\right.) ] contains a (k)-free number. This result improves on a previous result of M. Filaseta (J. Number Theory 30, No. 2 (1988), 208 225). ' 1995 Academic Press. Inc.

πŸ“œ SIMILAR VOLUMES

Some new results on k-free numbers
Some new results on k-free numbers
✍ Zaizhao Meng πŸ“‚ Article πŸ“… 2006 πŸ› Elsevier Science 🌐 English βš– 169 KB

In this paper we obtain an improved asymptotic formula on the frequency of k-free numbers with a given difference. We also give a new upper bound of Barban-Davenport-Halberstam type for the k-free numbers in arithmetic progressions.

On the independence number in K1, r+1-fr
On the independence number in K1, r+1-free graphs
✍ ZdenΔ›k RyjÑček; Ingo Schiermeyer πŸ“‚ Article πŸ“… 1995 πŸ› Elsevier Science 🌐 English βš– 353 KB

In this paper we use the degree sequence, order, size and vertex connectivity of a K 1,,+ 1 -free graph or of an almost claw-free graph to obtain several upper bounds on its independence number. We also discuss the sharpness of these results.