A136259 Stone skipping numbers.

1, 3, 4, 5, 9, 13, 18, 19, 31, 32, 33, 38, 39, 55, 56, 57, 58, 59, 94, 95, 96, 97, 103, 104, 156, 157, 239, 244, 245, 249, 253, 254, 255, 256, 257, 258, 275, 276, 277, 419, 420, 609, 610, 787, 788, 789, 790, 791, 792, 1069, 1070, 1664, 1665, 1666, 1667, 1668, 1669, 1670

Offset

1,2

Comments

The sequence is generated by a sieving method with iterated selection of intervals of the natural numbers as if they were forming a chain of contact points on which a stone could re-bounce once launched at some specific position at the small numbers.

Image a stone with an initial kinetic energy t, which is diminished/dissipated by 1 unit each time it rebounds from the "water surface" of the residual sequence; it rebounds t times and sinks once it has slowed down to t=1. The numbers underneath the arcs of this flight, but not the contact points, are eliminated. We look at the limit of repeatedly skipping stones each time starting at new launching points with larger initial t. In detail:

Start with the set of natural numbers. Let a(0)= t define t. Jump t positions to the right, erase t positions; from the last erased position jump t-1 positions to the right, erase t-1 positions; ...; jump 1 position to the right, erase 1 position. Go to the smallest i>t. Set t=i. Repeat.

Stone skipping sequences are a generalized case of scarce sequences; see A137292.

Links

Table of n, a(n) for n=1..58.

L. Bocquet, The physics of stone skipping, Am. J. Phys 71 (2) (2003) 150-155.

D. X. Charles, Sieve Methods, July 2000, U. of Wisconsin.

Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT], 2007-2010.

M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI, 1969, pp.41-56.

Example

Start with natural numbers
   1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,...
a(0)=1 set t=1 (jump 1 position to the right, erase 1 position) gives
   1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,...
i=3 set t=3 (jump 3 positions to the right, erase 3 positions; from the last erased position jump 2 positions to the right, erase 2 positions; from the last erased position jump 1 position to the right, erase 1 position) gives
   1,3,4,5,9,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,...
i=4 set t=4 (jump 4 positions to the right, erase 4 positions; from the last erased position jump 3 positions to the right, erase 3 positions; from the last erased position jump 2 positions to the right, erase 2 positions;from the last erased position jump 1 position to the right, erase 1 position ) gives
   1,3,4,5,9,13,18,19,23,27,28,...
i=5 set t=5, repeat procedure.

Maple

nmax := 3000: a136259 := [seq(i, i=1..nmax)] : s := 1: t := op(s, a136259) : p := 1:
while op(-1, a136259)>t do p := p+t ; outb := false; while t >= 1 do for eli from 1 to t do if p > nops(a136259) then outb := true; break; fi; a136259 := subsop(p=NULL, a136259) ; od: if outb then break; fi; t := t-1 ; p := p+t-1 ; od: print(a136259) ; s := s+1 ; p := s ; t := op(s, a136259) : od: # R. J. Mathar, Aug 17 2009

Crossrefs

Cf. A137292. Bisections are A238091, A238092.

Cf. A270877.

Sequence in context: A080633 A360374 A242800 * A099560 A356604 A050161

Adjacent sequences: A136256 A136257 A136258 * A136260 A136261 A136262

Keyword

easy,nonn

Author

Ctibor O. Zizka, Mar 18 2008

Extensions

Edited and corrected by R. J. Mathar, Aug 17 2009

Status

approved