The Josephus problem
Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, p. 303-318
Nous donnons des formules explicites permettant de calculer les nombres de Josephus j(n,2,i) and j(n,3,i) et fournissant une majoration et une minoration explicites de j(n,k,i) qui ne diffèrent que d’au plus 2k-2 (dans le cas k=4, ces bornes sont même meilleures). Nous proposons aussi un nouvel algorithme pour le calcul de ces nombres basé précisément sur ces estimations.
We give explicit non-recursive formulas to compute the Josephus-numbers j(n,2,i) and j(n,3,i) and explicit upper and lower bounds for j(n,k,i) (where k4) which differ by 2k-2 (for k=4 the bounds are even better). Furthermore we present a new fast algorithm to calculate j(n,k,i) which is based upon the mentioned bounds.
@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},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     year = {1997},
     pages = {303-318},
     zbl = {0905.05002},
     mrnumber = {1617400},
     language = {en},
     url = {http://http://www.numdam.org/item/JTNB_1997__9_2_303_0}
}
Halbeisen, Lorenz; Hungerbühler, Norbert. The Josephus problem. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 303-318. http://www.numdam.org/item/JTNB_1997__9_2_303_0/

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