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A321781 Least q > 1 letting Josephus survive if he finds himself at position j in the circle of m persons, but is allowed to name the elimination parameter q such that every q-th person is executed, written as triangle T(m,j), m > 1, j <= m.

%I A321781 #16 Jun 23 2020 10:18:58
%S A321781 0,2,3,5,3,2,2,4,6,10,4,5,2,3,11,3,10,8,6,2,27,11,4,6,3,7,5,2,2,19,5,
%T A321781 7,12,4,3,9,3,7,2,42,35,11,6,5,21,8,19,5,3,2,15,9,10,7,12,16,26,24,40,
%U A321781 7,36,2,5,4,14,12,4,9,6,26,8,11,18,13,2,3,12,7,21,10,15,11,4,5,23,13,6,12,2,18,3
%N A321781 Least q > 1 letting Josephus survive if he finds himself at position j in the circle of m persons, but is allowed to name the elimination parameter q such that every q-th person is executed, written as triangle T(m,j), m > 1, j <= m.
%C A321781 Exercise 23 associated with Chapter 1.3 in "Concrete Mathematics" about the Josephus Problem asks: "Suppose that Josephus finds himself in a given position j, but he has a chance to name the elimination parameter q such that every qth person is executed. Can he always save himself?"
%C A321781 T(1,1) is set to 0 to complete the triangle. q > 1 serves to avoid the obviously merciless choice of q = 1 in the case of Josephus being located at position m.
%D A321781 Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994, page 20.
%H A321781 <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%e A321781 The triangle begins:
%e A321781    0
%e A321781    2  3
%e A321781    5  3  2
%e A321781    2  4  6 10
%e A321781    4  5  2  3 11
%e A321781    3 10  8  6  2 27
%e A321781   11  4  6  3  7  5  2
%e A321781    2 19  5  7 12  4  3  9
%e A321781    3  7  2 42 35 11  6  5 21
%e A321781    8 19  5  3  2 15  9 10  7 12
%e A321781   16 26 24 40  7 36  2  5  4 14 12
%e A321781    4  9  6 26  8 11 18 13  2  3 12  7
%e A321781    ...
%e A321781 3 persons:
%e A321781   q = 2: 111 -> 101 -> 001. Position 3 survives, therefore T(3,3) = 2;
%e A321781   q = 3: 111 -> 110 -> 010. Position 2 survives, therefore T(3,2) = 3;
%e A321781   q = 4: 111 -> 011 -> 010. Position 2 survives, already covered by q = 3;
%e A321781   q = 5: 111 -> 101 -> 100. Position 1 survives, therefore T(3,1) = 5.
%Y A321781 The first column of the table is A187788.
%Y A321781 Cf. A003418, A032434, A321793, A321794.
%K A321781 nonn,tabl
%O A321781 1,2
%A A321781 _Hugo Pfoertner_, Nov 18 2018