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A357217 Array read by descending antidiagonals: T(n,k) is the number of cycles of the permutation given by the order of elimination in the Josephus problem for n numbers and a count of k; n, k >= 1.

%I A357217 #9 Sep 18 2022 11:22:15
%S A357217 1,1,2,1,1,3,1,2,2,4,1,1,1,2,5,1,2,2,2,1,6,1,1,1,2,1,1,7,1,2,2,2,1,2,
%T A357217 4,8,1,1,3,2,3,3,3,2,9,1,2,2,2,3,2,2,2,1,10,1,1,1,2,1,3,3,2,3,5,11,1,
%U A357217 2,2,2,3,2,2,4,5,2,2,12,1,1,1,2,3,1,3,2,3,1,3,2,13
%N A357217 Array read by descending antidiagonals: T(n,k) is the number of cycles of the permutation given by the order of elimination in the Josephus problem for n numbers and a count of k; n, k >= 1.
%C A357217 n >= 2 is a Josephus_k prime if and only if T(n,k) = 1; see A163782-A163800.
%H A357217 Pontus von Brömssen, <a href="/A357217/b357217.txt">Antidiagonals n = 1..100, flattened</a>
%H A357217 James Dowdy and Michael E. Mays, <a href="https://www.researchgate.net/publication/266756060">Josephus permutations</a>, Journal of Combinatorial Mathematics and Combinatorial Computing 6 (1989), 125-130.
%H A357217 Wikipedia, <a href="https://en.wikipedia.org/wiki/Josephus_problem">Josephus problem</a>
%H A357217 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%F A357217 T(n,k+A003418(n)) = T(n,k), i.e., the n-th row is periodic with period A003418(n).
%e A357217 Array begins:
%e A357217   n\k|  1  2  3  4  5  6  7  8  9 10
%e A357217   ---+------------------------------
%e A357217    1 |  1  1  1  1  1  1  1  1  1  1
%e A357217    2 |  2  1  2  1  2  1  2  1  2  1
%e A357217    3 |  3  2  1  2  1  2  3  2  1  2
%e A357217    4 |  4  2  2  2  2  2  2  2  2  2
%e A357217    5 |  5  1  1  1  3  3  1  3  3  3
%e A357217    6 |  6  1  2  3  2  3  2  1  2  3
%e A357217    7 |  7  4  3  2  3  2  3  2  5  2
%e A357217    8 |  8  2  2  2  4  2  2  4  6  2
%e A357217    9 |  9  1  3  5  3  3  3  3  3  3
%e A357217   10 | 10  5  2  1  2  3  2  1  2  3
%e A357217 For n = 4, k = 2, the order of elimination is (2,4,3,1) (row 4 of A321298). This permutation has two cycles, (1 2 4) and (3), so T(4,2) = 2.
%o A357217 (Python)
%o A357217 from sympy.combinatorics import Permutation
%o A357217 def A357217(n,k):
%o A357217     return Permutation.josephus(k,n).cycles
%Y A357217 Cf. A003418, A006694 (column k=2), A163782-A163800 (Josephus primes), A198789, A321298 (the Josephus permutations for k=2).
%K A357217 nonn,tabl
%O A357217 1,3
%A A357217 _Pontus von Brömssen_, Sep 18 2022