OFFSET
0,3
COMMENTS
Solution to the classic "Monkey and Coconut Problem" for n sailors.
Also called "Sailors and Monkey Problem": a(n) is smallest number such that C -> (C-1)*(1-1/n) can be applied n times and at every step have an integer C == 1 (mod n).
The expression for a(n) is easily derived from the observation that had an extra n-1 coconuts been added to the original pile a(n), the monkey would have been doomed to a zero coconut tip all through, the successive heaps of leftovers then collapsing to an ordinary geometric progression of common ratio (1 - 1/n). For a total number of (n+1) interventions, we thus have n^(n+1) dividing a(n) + (n-1), whence the formula. - Lekraj Beedassy, Jun 04 2002
REFERENCES
H. E. Dudeney, The Canterbury Puzzles, Prob. 114 pp. 160-161, 250, Nelson, London 1919.
M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles & Diversions, p. 108, Simon & Shuster, NY 1961.
P. Halmos, Problems for Mathematicians Young and Old, MAA DC 1991.
W. L. Schaff, A Bibliography of Recreational Mathematics, Vol. 2 Chap. 1.18c, p. 24, NCTM Washington D. C., 1970.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
Anonymous, The Coconut Puzzle.
J. Burkardt, The Coconut puzzle.
Santo D'Agostino, "The Coconut Problem"; Updated With Solution, May 2011.
R. V. Gassel et al., Coconut Chaos.
MathKnox, Puzzle of the week.
J. S. Tanton, A collection of research problems.
K. Uhland, Marx Brothers, Four Years Later.
K. Uhland, Marx Brothers, Four Years Later.
Eric Weisstein's World of Mathematics, Monkey and Coconut Problem.
FORMULA
E.g.f.: e^x*(1-x) + T/(1-T)^3, where T=T(x) is Euler's tree function (see A000169). - Len Smiley Dec 10 2001
MATHEMATICA
Table[n^(n+1)-n+1, {n, 0, 30}] (* Harvey P. Dale, Mar 24 2011 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Additional links supplied by Lekraj Beedassy
STATUS
approved