A014293 a(n) = n^(n+1) - n + 1.

1, 1, 7, 79, 1021, 15621, 279931, 5764795, 134217721, 3486784393, 99999999991, 3138428376711, 106993205379061, 3937376385699277, 155568095557812211, 6568408355712890611, 295147905179352825841, 14063084452067724990993

Offset: 0

Author

Russ Cox

Keywords

• nonn
• easy
• nice

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.

Programs

Mathematica

Table[n^(n+1)-n+1,{n,0,30}]  (* Harvey P. Dale, Mar 24 2011 *)

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

Extensions

Additional links supplied by Lekraj Beedassy