A137292 -id:A137292 - cp's OEIS frontend

A138662 First term of the first run of at least n consecutive numbers in A137292.

1, 3, 3, 469, 1333, 15984, 978131, 978131

Offset: 1

Klaus Brockhaus, Mar 25 2008

			First few terms of A137292 are 1,3,4,5,7,9,10,13,... . The first run of at least two consecutive numbers is 3,4; the first run of at least three consecutive numbers is 3,4,5. Hence a(2) = a(3) = 3.

Cf. A137292.

A138845 First gap of length at least n in A137292, lower end.

Original entry on oeis.org

1, 10, 115, 201, 1015, 1029, 10029, 100021

Offset: 1

Klaus Brockhaus, Mar 31 2008

			First few terms of A137292 are 1,3,4,5,7,8,10,13,14,15,17,... . The first gap of length 1 is between 1 and 3, hence a(1) = 1; the first gap of length 2 is between 10 and 13, hence a(2) = 10.

Cf. A137292, A138662.

A136119 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + a(n).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 71, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 95, 96, 97, 99, 100

Offset: 1

Ctibor O. Zizka, Mar 16 2008

Apparently a(n) = A001953(n-1)+1 = floor((n-1/2)*sqrt(2))+1 (confirmed for n < 20000) and a(n+1) - a(n) = A001030(n). From the definitions these conjectures are by no means obvious. Can they be proved? - Klaus Brockhaus, Apr 15 2008 [For an affirmative answer, see the Cloitre link.]

This is the s(n)-Wythoff sequence for s(n)=2n-1; see A184117 for the definition. Complement of A184119. - Clark Kimberling, Jan 09 2011

			First few steps are:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 1; delete term at position 1+a(1) = 2: 2;
1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 2; delete term at position 2+a(2) = 5: 6;
1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 3; delete term at position 3+a(3) = 7: 9;
1,3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,20,...
n = 4; delete term at position 4+a(4) = 9: 12;
1,3,4,5,7,8,10,11,13,14,15,16,17,18,19,20,...
n = 5; delete term at position 5+a(5) = 12: 16;
1,3,4,5,7,8,10,11,13,14,15,17,18,19,20,...
n = 6; delete term at position 6+a(6) = 14: 19;
1,3,4,5,7,8,10,11,13,14,15,17,18,20,...

B. Cloitre, The golden sieve, preprint 2008

Alois P. Heinz, Table of n, a(n) for n = 1..1000
D. X. Charles, Sieve Methods, July 2000, University of Wisconsin.
Benoit Cloitre, On the proof of Klaus Brockhaus's conjectures
R. Eismann, Decomposition of natural numbers into weight X level + jump and application to a new classification of prime numbers, arXiv:0711.0865 [math.NT], 2007-2010.
M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp.41-56.
Index entries for sequences generated by sieves

Cf. A000027, A001953 (floor((n+1/2)*sqrt(2))), A001030 (fixed under 1 -> 21, 2 -> 211), A136110, A137292.
Cf. A000959, A099267.
Cf. A242535.
Cf. A000217 (T).

import Data.List (delete)
a136119 n = a136119_list !! (n-1)
a136119_list = f [1..] where
   f zs@(y:xs) = y : f (delete (zs !! y) xs)
-- Reinhard Zumkeller, May 17 2014

[Ceiling((n-1/2)*Sqrt(2)): n in [1..100]]; // Vincenzo Librandi, Jul 01 2019

f[0] = Range[100]; f[n_] := f[n] = Module[{pos = n + f[n-1][[n]]}, If[pos > Length[f[n-1]], f[n-1], Delete[f[n-1], pos]]]; f[1]; f[n = 2]; While[f[n] != f[n-1], n++]; f[n] (* Jean-François Alcover, May 08 2019 *)
T[n_] := n (n + 1)/2; Table[1 + 2 Sqrt[T[n-1]] , {n, 1, 71}] // Floor (* Ralf Steiner, Oct 23 2019 *)

apply( {A136119(n)=sqrtint(n*(n-1)*2)+1}, [1..99]) \\ M. F. Hasler, Jul 04 2022

a(n) = ceiling((n-1/2)*sqrt(2)). This can be proved in the same way as the formula given for A099267. There are some generalizations. For instance, it is possible to consider "a(n)+K*n" instead of "a(n)+n" for deleting terms where K=0,1,2,... is fixed. The constant involved in the Beatty sequence for the sequence of deleted terms then depends on K and equals (K + 1 + sqrt((K+1)^2 + 4))/2. K=0 is related to A099267. 1+A001954 is the complement sequence of this sequence A136119. - Benoit Cloitre, Apr 18 2008
a(n) = floor(1 + 2*sqrt(T(n-1))), with triangular numbers T(). - Ralf Steiner, Oct 23 2019
Lim_{n->inf}(a(n)/(n - 1)) = sqrt(2), with {a(n)/(n - 1)} decreasing. - Ralf Steiner, Oct 24 2019

Edited and extended by Klaus Brockhaus, Apr 15 2008

An incorrect g.f. removed by Alois P. Heinz, Dec 14 2012

A136259 Stone skipping numbers.

Original entry on oeis.org

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

Ctibor O. Zizka, Mar 18 2008

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.

			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.

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.

Cf. A137292. Bisections are A238091, A238092.

Cf. A270877.

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

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

A136272 Waterbird take-off sequence. Complement of A166021.

Original entry on oeis.org

1, 3, 6, 7, 11, 12, 13, 18, 19, 20, 21, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 43, 51, 52, 53, 54, 55, 56, 57, 66, 67, 68, 69, 70, 71, 72, 73, 83, 84, 85, 86, 87, 88, 89, 90, 91, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 123, 124, 125, 126, 127, 128, 129

Offset: 1

Ctibor O. Zizka, Mar 19 2008

This kind of sequence can be generalized as follows:
Let F(t), G(t) be arithmetic functions: F(t) the right hand move, G(t) the number of erased positions.
Then starting from the position t=1 do procedure:
JUMP F(t) positions right hand
ERASE G(t) positions
SET t=t+1
repeat procedure from the last erased position.
This sequence has F(t)=t, G(t)=t.
We can use a set of functions F_i(t) and G_i(t) processed in parallel (a flock of birds taking off).

			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,...
t=1; from the position 1 go 1 position to the right, erase 1 position:
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,...
t=2; from the last erased position go 2 positions to the right, erase 2 positions:
1..3..,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
t=3; from the last erased position go 3 positions to the right, erase 3 positions:
1..3..,6,7....11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,...
t=4; from the last erased position go 4 positions to the right, erase 4 positions:
1..3..,6,7....11,12,13....,18,19,20,21,22,23,24,25,26,27,...
t=5; from the last erased position go 5 positions to the right, erase 5 positions:
1..3..,6,7....11,12,13....,18,19,20,21......27,...
The erased positions form the complement of this sequence: A166021.

D. X. Charles, Sieve Methods, July 2000, University of Wisconsin.
R. Eismann, Decomposition of natural numbers into weight X level + jump and application to a new classification of prime numbers, arXiv:0711.0865 [math.NT], 2007-2010.
M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp. 41-56.

Cf. A004202, A064801, A137292.

is_A136272(n) = if(n<3, n==1, my(s=sqrtint(n-2)); n-2 < s*(s+1)) \\ Hugo Pfoertner, Jul 19 2024

a(0)=1; let t=1. Start on position t. Jump t positions right hand. Erase t positions. (*P*) Set t=t+1. Start on the last erased position. Jump t positions right hand. Erase t positions. Repeat procedure (*P*).

Edited and corrected by Antti Karttunen, Oct 05 2009

A136120 Limiting sequence when we start with the positive integers (A000027) and at step n >= 1 delete the a(n) terms at positions n+a(n) to n-1+2*a(n).

Original entry on oeis.org

1, 3, 4, 5, 9, 10, 15, 16, 22, 23, 24, 25, 26, 36, 37, 48, 49, 50, 51, 52, 53, 69, 70, 87, 88, 89, 90, 91, 92, 93, 116, 117, 141, 142, 167, 168, 194, 195, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 269, 270, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317

Offset: 1

Ctibor O. Zizka, Mar 16 2008

			First few steps are:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,...
n = 1, a(1) = 1; delete terms at positions 2 to 2; this is 2;
1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,...
n = 2,a(2) = 3; delete terms at positions 5 to 7; these are 6,7,8;
1,3,4,5,9,10,11,12,13,14,15,16,17,18,19,20,21,22,...
n = 3, a(3) = 4; delete terms at positions 7 to 10; these are 11,12,13,14;
1,3,4,5,9,10,15,16,17,18,19,20,21,22,23,24,25,26,27,28,...
n = 4, a(4) = 5; delete terms at positions 9 to 13; these are 17,18,19,20,21;
1,3,4,5,9,10,15,16,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36...
n = 5 a(5) = 9; delete terms at positions 14 to 22; these are 27,28,29,30,31,32,33,34,35;
1,3,4,5,9,10,15,16,22,23,24,25,26,36,...

Klaus Brockhaus, Table of n, a(n) for n=1..10000
D. X. Charles, Sieve Methods, July 2000, University of Wisconsin.
Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT]
M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp.41-56.
Index entries for sequences generated by sieves

Cf. A000027, A139418, A139419, A139420, A136110, A136119, A137292.

f[seq_] := Module[{s = seq, n1, n2}, n++; n1 = s[[n]] + n; If[n1 <= len, n2 = Min[n - 1 + 2*s[[n]], len]; len -= n2 - n1 + 1; Drop[s, {n1, n2}], s]]; n = 0; len = 1000; FixedPoint[f, Range[len]] (* Jean-François Alcover, Sep 29 2011 *)

Edited and extended by Klaus Brockhaus, Apr 20 2008

A136110 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + tau(a(n)), where tau(k) is the number of divisors of k.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 12, 13, 15, 17, 18, 22, 23, 24, 28, 29, 30, 32, 33, 36, 37, 38, 43, 44, 47, 49, 51, 52, 55, 56, 58, 59, 62, 65, 66, 68, 70, 72, 73, 74, 78, 79, 80, 84, 85, 86, 88, 90, 92, 94, 96, 97, 98, 104, 105, 106, 108, 109, 111, 116, 118, 119, 121, 122, 126, 129, 130

Offset: 1

Ctibor O. Zizka, Mar 16 2008

			First few steps are:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 1; delete term at position 1+tau(1) = 1+1 =2: 2;
1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 2; delete term at position 2+tau(3) = 1+2 = 3: 5;
1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 3; delete term at position 3+tau(4) = 3+3 = 6: 8;
1,3,4,6,7,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 4; delete term at position 4+tau(6) = 4+4 = 8: 11;
1,3,4,6,7,9,10,12,13,14,15,16,17,18,19,20,...
n = 5; delete term at position 5+tau(7) = 5+2 = 7: 10;
1,3,4,6,7,9,12,13,14,15,16,17,18,19,20,...
n = 6; delete term at position 6+tau(9) = 6+3 = 9: 14;
1,3,4,6,7,9,12,13,15,16,17,18,19,20,...

D. X. Charles, Sieve Methods, July 2000, University of Wisconsin.
Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT]
M. C. Wunderlich, A general class of sieve generated sequences, Acta Arithmetica XVI,1969, pp.41-56.
Index entries for sequences generated by sieves

Cf. A000005 (number of divisors), A000027, A137292, A138899, A138900.

Edited and extended by Klaus Brockhaus, Apr 03 2008

Moved references to the Links section R. J. Mathar, Oct 23 2009

A137832 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the last digit in the term at position n + a(n).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 10, 11, 1, 1, 1, 16, 1, 1, 1, 21, 2, 23, 2, 25, 2, 27, 2, 29, 3, 31, 3, 3, 34, 3, 3, 37, 3, 3, 40, 4, 43, 44, 45, 47, 48, 49, 5, 51, 52, 53, 5, 5, 56, 57, 5, 5, 6, 61, 6, 6, 6, 65, 6, 67, 69, 7, 7, 72, 7, 7, 75, 76, 7, 7, 79, 8, 82, 83, 84, 8

Offset: 1

Ctibor O. Zizka, Apr 29 2008

			First few steps are:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 1; delete the last digit in the term at position 1+a(1) = 2: 2;
1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 2; delete the last digit in the term at position 2+a(2) = 5: 6;
1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 3; delete the last digit in the term at position 3+a(3) = 7: 9;
1,3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,20,...
n = 4; delete the last digit in the term at position 4+a(4) = 9: 2;
1,3,4,5,7,8,10,11,1,13,14,15,16,17,18,19,20,...
n = 5; delete the last digit in the term at position 5+a(5) = 12: 5;
1,3,4,5,7,8,10,11,1,13,14,1,16,17,18,19,20,...
n = 6; delete the last digit in the term at position 6+a(6) = 14: 8;
1,3,4,5,7,8,10,11,1,13,14,1,16,17,1,19,20,...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A137292, A136120, A136119, A136259, A136272, A137319, A137417, A137418.

A137838 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the first digit in the term at position n + a(n).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 10, 11, 2, 13, 4, 5, 16, 7, 8, 19, 21, 2, 23, 4, 5, 6, 7, 8, 29, 0

Offset: 1

Ctibor O. Zizka, Apr 29 2008

			First few steps are:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 1; delete the first digit in the term at position 1+a(1) = 2: 2;
1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 2; delete the first digit in the term at position 2+a(2) = 5: 6;
1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,...
n = 3; delete the first digit in the term at position 3+a(3) = 7: 9;
1,3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,20,...
n = 4; delete the first digit in the term at position 4+a(4) = 9: 1;
1,3,4,5,7,8,10,11,2,13,14,15,16,17,18,19,20,...
n = 5; delete the first digit in the term at position 5+a(5) = 12: 1;
1,3,4,5,7,8,10,11,2,13,14,5,16,17,18,19,20,...
n = 6; delete the first digit in the term at position 6+a(6) = 14: 1;
1,3,4,5,7,8,10,11,2,13,14,5,16,7,18,19,20,...

Cf. A137292, A136120, A136119, A136259, A136272, A137319, A137417, A137418.

A138095 Limiting sequence when we start with positive integers (A000027) and at step n >= 1 add: to the term at position n + 1a(n) the value 1, to the term at position n + 2a(n) the value 1, ..., to the term at position n + n*a(n) the value 1.

Original entry on oeis.org

1, 3, 3, 4, 6, 7, 7, 10, 10, 10, 12, 14, 14, 15, 15, 17, 18, 19, 20, 23, 22, 22, 25, 24, 25, 27, 29, 30, 32, 32, 31, 32, 34, 35, 39, 36, 38, 39, 41, 42, 43, 43, 45, 46, 46, 46, 48, 52, 51, 53, 51, 52, 55, 55, 56, 59, 57, 60, 63, 62, 62, 64, 63, 65, 66, 68, 69, 70, 72, 71

Offset: 1

Ctibor O. Zizka, May 03 2008

			        n | 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  --------+-----------------------------------------------------------
  Step  1 |    3
  Step  2 |             6        9
  Step  3 |                7       10       13
  Step  4 |                     10          14          17          21
  Step  5 |                              12                18
  Step  6 |                                    14                   22
  Step  7 |                                       15
  Step  8 |                                                   19
  Step  9 |                                                      20
  Step 10 |                                                         23
  --------+-----------------------------------------------------------
     a(n) | 1  3  3  4  6  7  7 10 10 10 12 14 14 15 15 17 18 19 20 23

Cf. A000027, A136119, A137319, A137417, A137418, A136259, A136272, A137292, A136120, A137832, A137838, A137894, A137898, A137901, A137902, A137903.

lista(nn) = my(va = [1..nn]); for (n=1, nn, for (k=1, n, my(j = n+k*va[n]); if (j <= #va, va[j]++); )); va; \\ Michel Marcus, Aug 09 2022

Corrected and extended by Michel Marcus, Aug 09 2022

cp's OEIS Frontend

A138662 First term of the first run of at least n consecutive numbers in A137292.

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A138845 First gap of length at least n in A137292, lower end.

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A136119 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + a(n).

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A136259 Stone skipping numbers.

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A136272 Waterbird take-off sequence. Complement of A166021.

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A136120 Limiting sequence when we start with the positive integers (A000027) and at step n >= 1 delete the a(n) terms at positions n+a(n) to n-1+2*a(n).

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A136110 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the term at position n + tau(a(n)), where tau(k) is the number of divisors of k.

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A137832 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the last digit in the term at position n + a(n).

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A137838 Limiting sequence when we start with the positive integers (A000027) and delete in step n >= 1 the first digit in the term at position n + a(n).

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A138095 Limiting sequence when we start with positive integers (A000027) and at step n >= 1 add: to the term at position n + 1a(n) the value 1, to the term at position n + 2a(n) the value 1, ..., to the term at position n + n*a(n) the value 1.