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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. 5
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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?"
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.
REFERENCES
Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994, page 20.
LINKS
EXAMPLE
The triangle begins:
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 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 7 36 2 5 4 14 12
4 9 6 26 8 11 18 13 2 3 12 7
...
3 persons:
q = 2: 111 -> 101 -> 001. Position 3 survives, therefore T(3,3) = 2;
q = 3: 111 -> 110 -> 010. Position 2 survives, therefore T(3,2) = 3;
q = 4: 111 -> 011 -> 010. Position 2 survives, already covered by q = 3;
q = 5: 111 -> 101 -> 100. Position 1 survives, therefore T(3,1) = 5.
CROSSREFS
The first column of the table is A187788.
Sequence in context: A202694 A123221 A197032 * A254862 A340641 A322235
KEYWORD
nonn,tabl
AUTHOR
Hugo Pfoertner, Nov 18 2018
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)