We give explicit non-recursive formulas to compute the Josephus-numbers and and explicit upper and lower bounds for (where ) which differ by (for the bounds are even better). Furthermore we present a new fast algorithm to calculate which is based upon the mentioned bounds.
Nous donnons des formules explicites permettant de calculer les nombres de Josephus and et fournissant une majoration et une minoration explicites de qui ne diffèrent que d’au plus (dans le cas , ces bornes sont même meilleures). Nous proposons aussi un nouvel algorithme pour le calcul de ces nombres basé précisément sur ces estimations.
@article{JTNB_1997__9_2_303_0, author = {Halbeisen, Lorenz and Hungerb\"uhler, Norbert}, title = {The {Josephus} problem}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {303--318}, publisher = {Universit\'e Bordeaux I}, volume = {9}, number = {2}, year = {1997}, mrnumber = {1617400}, zbl = {0905.05002}, language = {en}, url = {http://archive.numdam.org/item/JTNB_1997__9_2_303_0/} }
Halbeisen, Lorenz; Hungerbühler, Norbert. The Josephus problem. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 2, pp. 303-318. http://archive.numdam.org/item/JTNB_1997__9_2_303_0/
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