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

Author

Ctibor O. Zizka, Mar 18 2008

Keywords

• easy
• nonn

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A137292. Bisections are A238091, A238092.

Cf. A270877.

Programs

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

Extensions

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