{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:42:58Z","timestamp":1753893778720,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A set $A\\subseteq {\\bf N}$ is $(k,\\ell)$-sum-free, for  $k,\\ell\\in {\\bf N}$, $k&gt;\\ell$, if it contains no solutions to the equation $x_1+\\dots+x_k=y_1+\\dots+y_{\\ell}$. Let $\\rho=\\rho (k-\\ell)$   be the smallest natural number not dividing $k-\\ell$, and let $r=r_n$,  $0\\le r  &lt; \\rho$,  be such that $r\\equiv n \\pmod {\\rho }$. The main result of this note says that if $(k-\\ell)\/\\ell$ is small in  terms of $\\rho$, then the number of $(k,\\ell)$-sum-free subsets of $[1,n]$ is equal to $(\\varphi(\\rho)+\\varphi_r(\\rho)+o(1)) 2^{\\lfloor n\/\\rho \\rfloor}$, where $\\varphi_r(x)$ denotes the number of positive integers $m\\le r$ relatively prime to $x$ and $\\varphi(x)=\\varphi_x(x)$.<\/jats:p>","DOI":"10.37236\/1508","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:04:34Z","timestamp":1578708274000},"source":"Crossref","is-referenced-by-count":4,"title":["A Note on the Number of $(k,l)$-Sum-Free Sets"],"prefix":"10.37236","volume":"7","author":[{"given":"Tomasz","family":"Schoen","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2000,5,23]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r30\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r30\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:23:03Z","timestamp":1579324983000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v7i1r30"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,5,23]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2000,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1508","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2000,5,23]]},"article-number":"R30"}}