# On a question of Eliahou and a conjecture of Wilf

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## Abstract

To a numerical semigroup *S*, Eliahou associated a number \({{\mathrm{E}}}(S)\) and proved that numerical semigroups for which the associated number is non negative satisfy Wilf’s conjecture. The search for counterexamples for the conjecture of Wilf is therefore reduced to semigroups which have an associated negative Eliahou number. Eliahou mentioned 5 numerical semigroups whose Eliahou number is \(-1\). The examples were discovered by Fromentin who observed that these are the only ones with negative Eliahou number among the over \(10^{13}\) numerical semigroups of genus up to 60. We prove here that for any integer *n* there are infinitely many numerical semigroups *S* such that \({{\mathrm{E}}}(S)=n\), by explicitly giving families of such semigroups. We prove that all the semigroups in these families satisfy Wilf’s conjecture, thus providing not previously known examples of semigroups for which the conjecture holds.

## Keywords

Numerical semigroup Wilf conjecture Computational mathematics## Mathematics Subject Classification

20M14 05A20 11B75 20-04## Notes

### Acknowledgements

Besides the formal acknowledgement to the Centre for Mathematics of the University of Porto (CMUP) made somewhere in this paper, I would like to thank CMUP for providing me the computational tools necessary to this kind of works. I would like also to thank Shalom Eliahou for his suggestions and comments on preliminary versions. Special thanks to Shalom for encouraging me to publish this work. The anonimous referee, who carefully read the manuscript and gave me the appropriate remarks, deserves also my special ackowledgement.

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