A008336 -id:A008336 - cp's OEIS frontend

A337486 Numbers k such that b(k+1) divides b(k), where b() is Recamán's multiplicative sequence A008336.

1, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 36, 38, 42, 44, 45, 46, 50, 52, 54, 58, 60, 66, 68, 70, 72, 78, 82, 84, 86, 90, 92, 93, 94, 95, 98, 99, 100, 104, 106, 110, 111, 114, 116, 118, 119, 122, 124, 126, 130, 132, 134, 135, 136, 142, 146, 147, 148, 150, 154, 156, 158, 161, 162, 164, 165, 166

Offset: 1

Scott R. Shannon, Aug 29 2020

nonn

The old definition was: 1, together with the numbers formed by removing the required prime factors to form the number from a set which is initially empty and that has primes added via the addition of the prime factors of numbers which cannot be created from those currently in the set. Start by trying to create the number 2.
Consider an initially empty set of primes whose numbers are used to create a given number where each time a number is created those prime factors are removed from the set. If a number cannot be created as all its required prime factors are not currently in the set then all the prime factors of that number are instead added to the set. Start by trying to create the number 2 followed by all other integers. This sequence list the numbers that are created.
For the first 1 million terms the largest gap between terms is 15, between a(88018) = 189648 and a(88019) = 189663. The 1 millionth term is created after the addition of the number 2123404. The prime set at that point has 114323 primes, with the maximum number of entries for a single prime being eleven, for 887.
Note that if after the creation of a number the entire set of primes is cleared then the numbers created are those in A109895.

			a(2) = 6. As there are no primes initially in the set 2,3,4,5 cannot be created and instead these numbers add three 2's, one 3 and one 5 to the set. As there is now one 2 and one 3 the number 6 = 2*3 can be created. After 6 is created the set of primes now contains 2,2,5.
a(3) = 10. After 7,8,9, none of which can be created from the prime set, the prime set contains 2,2,2,2,2,3,3,5,7. As 2 and 5 are present 10 = 2*5 can be created, after which the set contains 2,2,2,2,3,3,7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005132, A008336, A331276, A330430, A109895.

First differences are A370969.

Block[{nn = 166, k = 1}, Reap[Do[If[Mod[k, i] == 0, k /= i; Sow[i], k *= i], {i, nn}]][[-1, 1]]] (* Michael De Vlieger, Sep 12 2020 *)

from itertools import count, islice
def A337486_gen(): # generator of terms
    c = 1
    for n in count(1):
        a, b = divmod(c,n)
        if not b:
            c = a
            yield n
        else:
            c *= n
A337486_list = list(islice(A337486_gen(),30)) # Chai Wah Wu, Apr 11 2024

A370968 A008336 sorted and duplicates removed.

Original entry on oeis.org

1, 2, 6, 20, 24, 120, 140, 858, 924, 1008, 1120, 10080, 11088, 12012, 12870, 176358, 184756, 194480, 205920, 3500640, 3695120, 3879876, 4056234, 87253605, 90262350, 93605400, 97349616, 2433740400, 2527345800, 2617608150, 2704861755, 57176602275, 79526843550, 81676217700, 84009823920, 86555576160, 2630123704650, 2687300306925, 2856334013280

Offset: 1

N. J. A. Sloane, Apr 11 2024

nonn

In fact 1 is the only repeated term in A008336 (see that entry for proof).

			From _Michael De Vlieger_, Apr 11 2024: (Start)
Table of first terms showing exponents of prime factors p | a(n), with "." signifying p^0.
                 primes
                     1111
   n       a(n)  23571379
  -----------------------
   1         1   .
   2         2   1
   3         6   11
   4        20   2.1
   5        24   31
   6       120   311
   7       140   2.11
   8       858   11..11
   9       924   21.11
  10      1008   42.1
  11      1120   5.11
  12     10080   5211
  13     11088   42.11
  14     12012   21.111
  15     12870   121.11
  16    176358   11.1.111
  17    184756   2...1111
  18    194480   4.1.111
  19    205920   521.11
  20   3500640   521.111 (End)

Michael De Vlieger, Table of n, a(n) for n = 1..2753
Michael De Vlieger, Plot p(i)^m(i) | a(n) at (x,y) = (n,i), n = 1..1024, 4X vertical exaggeration, with a color function where m(i) = 1 is black, m(i) = 2 is red, ... largest m(i) for n <= 1024 is in magenta.

Cf. A008336, A337486.

A371908 a(n) = 2-adic valuation of A008336(2*n).

Original entry on oeis.org

0, 1, 3, 2, 5, 4, 2, 1, 5, 4, 2, 1, 4, 3, 1, 0, 5, 4, 2, 1, 4, 3, 1, 0, 4, 3, 1, 0, 3, 2, 0, 1, 7, 6, 4, 3, 0, 1, 3, 2, 6, 5, 3, 2, 5, 4, 2, 1, 6, 5, 3, 4, 1, 0, 2, 1, 5, 4, 2, 1, 4, 3, 1, 0, 7, 6, 4, 3, 0, 1, 3, 2, 6, 5, 3, 2, 5, 4, 2, 1, 6, 5, 3, 2, 5, 4, 2

Offset: 1

Michael De Vlieger, Apr 11 2024

Aside from initial 0, first 50 terms agree with A371905: A371905(50) = 3 while a(51) = 5.

			Let b(n) = A008336(n) and let f(x) = A007814(x).
a(1) = 0 since b(2*1) = 1 and f(b(2)) = 0.
a(2) = 1 since b(2*2) = 6 and f(b(4)) = 1.
a(3) = 3 since b(2*3) = 120 and f(b(6)) = 3, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007814, A008336, A371905, A372095, A372840.

k = 1; nn = 240; p[_] := 0; r = 0; q = Prime[k];
{0}~Join~Reap[
    Do[If[AnyTrue[#, p[#1] < #2 & @@ # &],
        Map[p[#1] += #2 & @@ # &, #],
        Map[p[#1] -= #2 & @@ # &, #] ] &@
        Map[{PrimePi[#1], #2} & @@ # &, FactorInteger[n]];
      If[Divisible[n, q], Sow[p[k] ] ], {n, nn}] ][[-1, 1]]

from itertools import count, islice
def A371908_gen(): # generator of terms
    m = 1
    for n in count(1,2):
        a, b = divmod(m,n)
        m = m*n if b else a
        yield (~m&m-1).bit_length()
        a, b = divmod(m,n+1)
        m = m*(n+1) if b else a
A371908_list = list(islice(A371908_gen(),20)) # Chai Wah Wu, Apr 15 2024

a(n) = A007814(A008336(2*n)).

A372095 a(n) = 3-adic valuation of A008336(3*n).

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 0, 1, 2, 5, 4, 5, 3, 4, 3, 1, 2, 3, 0, 1, 0, 2, 1, 2, 0, 1, 0, 4, 3, 4, 2, 1, 2, 0, 1, 2, 5, 4, 3, 5, 6, 7, 5, 6, 5, 2, 3, 4, 6, 5, 4, 6, 5, 6, 2, 1, 2, 0, 1, 2, 0, 1, 0, 3, 4, 5, 3, 4, 3, 5, 4, 5, 2, 3, 2, 0, 1, 0, 2, 3, 4, 9, 8, 9, 7, 8, 7

Offset: 1

Michael De Vlieger, Jun 02 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007949, A008336, A371908, A372840.

k = 2; nn = 360; p[_] := 0; r = 0; q = Prime[k];
{0}~Join~Reap[
   Do[If[AnyTrue[#, p[#1] < #2 & @@ # &],
        Map[p[#1] += #2 & @@ # &, #],
        Map[p[#1] -= #2 & @@ # &, #] ] &@
        Map[{PrimePi[#1], #2} & @@ # &, FactorInteger[n]];
      If[Divisible[n, q], Sow[p[k] ] ], {n, nn}] ][[-1, 1]]

a(n) = A007949(A008336(3*n)).

A372840 a(n) = 5-adic valuation of A008336(5*n).

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 1, 2, 3, 2, 0, 1, 0, 1, 0, 2, 3, 4, 3, 2, 0, 1, 0, 1, 2, 5, 4, 3, 4, 5, 3, 4, 5, 4, 3, 1, 0, 1, 0, 1, 3, 4, 3, 4, 5, 3, 2, 1, 2, 3, 0, 1, 0, 1, 0, 2, 3, 4, 3, 4, 2, 3, 2, 3, 4, 2, 1, 0, 1, 2, 0, 1, 0, 1, 0, 3, 4, 3, 2, 1, 3, 2, 1, 0, 1, 3, 2

Offset: 1

Michael De Vlieger, Jun 02 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A008336, A112765, A371908, A372095.

k = 3; nn = 600; p[_] := 0; r = 0; q = Prime[k];
{0}~Join~Reap[
    Do[If[AnyTrue[#, p[#1] < #2 & @@ # &],
        Map[p[#1] += #2 & @@ # &, #],
        Map[p[#1] -= #2 & @@ # &, #] ] &@
        Map[{PrimePi[#1], #2} & @@ # &, FactorInteger[n]];
      If[Divisible[n, q], Sow[p[k] ] ], {n, nn}] ][[-1, 1]]

a(n) = A112765(A008336(5*n)).

A195504 Product of numbers up to n-1 used as divisors in A008336(n), n >= 2; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 60, 60, 720, 720, 10080, 10080, 10080, 10080, 181440, 181440, 3628800, 3628800, 79833600, 79833600, 79833600, 79833600, 2075673600, 2075673600, 58118860800, 58118860800, 1743565824000, 1743565824000, 1743565824000, 1743565824000

Offset: 1

Daniel Forgues, Sep 19 2011

This sequence provides more insight into the asymptotic behavior of log(A008336(n)). - Daniel Forgues, Sep 21 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..500

Cf. A008336, A000142.

A008336 := proc(n) option remember; if(n=1)then 1: elif(procname(n-1) mod (n-1)=0)then procname(n-1)/(n-1): else procname(n-1)*(n-1): fi: end: A195504 := proc(n) option remember: if(n=1)then 1: elif(A008336(n)<A008336(n-1))then (n-1)*procname(n-1): else procname(n-1): fi: end: seq(A195504(n),n=1..40); # Nathaniel Johnston, Sep 29 2011

Sqrt((n-1)! / A008336(n)), n >= 1.

A369407 A variant of A008336 based on polynomials over GF(2) (see Comments for precise definition).

Original entry on oeis.org

1, 2, 6, 24, 120, 20, 108, 864, 96, 960, 6720, 624, 6192, 37152, 491232, 30702, 1806, 127, 1905, 27348, 486596, 25102, 1890, 19760, 456624, 5581280, 439712, 21624, 451032, 5199760, 123954032, 3966529024, 123317760, 3850804224, 127210628096, 4070965504

Offset: 1

Rémy Sigrist, Jan 22 2024

Let P(m) denote the polynomial over GF(2) whose coefficients are encoded in the binary expansion of the nonnegative integer m.
Let b(1) = 1 and for any n > 0, if P(n) divides b(n) then b(n+1) = b(n) / P(n), otherwise b(n+1) = b(n) * P(n).
For any n > 0, a(n) is the unique number v such that P(v) = b(n).

			The first terms, alongside the corresponding polynomials, are:
  n   a(n)  b(n)                   P(n)
  --  ----  ---------------------  -----------
   1     1  1                      1
   2     2  X                      X
   3     6  X^2 + X                X + 1
   4    24  X^4 + X^3              X^2
   5   120  X^6 + X^5 + X^4 + X^3  X^2 + 1
   6    20  X^4 + X^2              X^2 + X
   7   108  X^6 + X^5 + X^3 + X^2  X^2 + X + 1
   8   864  X^9 + X^8 + X^6 + X^5  X^3
   9    96  X^6 + X^5              X^3 + 1
  10   960  X^9 + X^8 + X^7 + X^6  X^3 + X

Rémy Sigrist, Table of n, a(n) for n = 1..3842
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A008336.

P(n) = Mod(1, 2) * Pol(binary(n))
P_1(p) = fromdigits(lift(Vec(p)), 2)
{ b = 1; for (n = 1, 36, p = P(n); if (b % p==0, b \= p, b *= p); print1 (P_1(b)", ");); }

A370971 A008336(n) is divisible by the product of the primes p such that n/2 <= p < n; a(n) is the quotient.

Original entry on oeis.org

1, 1, 1, 1, 8, 8, 4, 4, 32, 288, 144, 144, 12, 12, 6, 90, 1440, 1440, 80, 80, 4, 84, 42, 42, 1008, 25200, 12600, 340200, 12150, 12150, 405, 405, 12960, 427680, 213840, 7484400, 207900, 207900, 103950, 4054050, 162162000, 162162000, 3861000, 3861000, 87750, 1950, 975, 975, 46800, 2293200, 45864, 2339064, 44982

Offset: 1

N. J. A. Sloane, Apr 13 2024

nonn

			For n= 7, A008336(7) = 20. The only prime with 3.5 <= p < 7 is 5, so a(7) = 20/5 = 4.
For n= 8, A008336(7) = 140. The only primes with 4 <= p < 8 are 5 and 7, so a(8) = 140/(5*7) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..2732

Cf. A008336, A055773 (note that A055773 has offset 0).

a(n) = A008336(n)/A055773(n-1).

A005132 Recamán's sequence (or Recaman's sequence): a(0) = 0; for n > 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155

Offset: 0

N. J. A. Sloane and Simon Plouffe, May 16 1991

The name "Recamán's sequence" is due to N. J. A. Sloane, not the author!
I conjecture that every number eventually appears - see A057167, A064227, A064228. - N. J. A. Sloane. That was written in 1991. Today I'm not so sure that every number appears. - N. J. A. Sloane, Feb 26 2017
As of Jan 25 2018, the first 13 missing numbers are 852655, 930058, 930557, 964420, 966052, 966727, 969194, 971330, 971626, 971866, 972275, 972827, 976367, ... For further information see the "Status Report" link. - Benjamin Chaffin, Jan 25 2018
From David W. Wilson, Jul 13 2009: (Start)
The sequence satisfies [1] a(n) >= 0, [2] |a(n)-a(n-1)| = n, and tries to avoid repeats by greedy choice of a(n) = a(n-1) -+ n.
This "wants" to be an injection on N = {0, 1, 2, ...}, as it attempts to avoid repeats by choice of a(n) = a(n-1) + n when a(n) = a(n-1) - n is a repeat.
Clearly, there are injections satisfying [1] and [2], e.g., a(n) = n(n+1)/2.
Is there a lexicographically earliest injection satisfying [1] and [2]? (End)
Answer: Yes, of course: The set of injections satisfying [1] and [2] is not empty, so there's a lexicographically least element. More concretely, it starts with the same 23 terms a(0..22) which are known to be minimal, but after a(22) = 41 it has to go on with a(23) = 41 + 23 = 64, since choosing "-" here necessarily yields a non-injective sequence. See A171884. - M. F. Hasler, Apr 01 2019
It appears that a(n) is also the value of "x" and "y" of the endpoint of the L-toothpick structure mentioned in A210606 after n-th stage. - Omar E. Pol, Mar 24 2012
Of course this is not a permutation of the integers: the first repeated term is 42 = a(24) = a(20). - M. F. Hasler, Nov 03 2014. Also 43 = a(18) = a(26). - Jon Perry, Nov 06 2014
Of all the sequences in the OEIS, this one is my favorite to listen to. Click the "listen" button at the top, set the instrument to "103. FX 7 (Echoes)", click "Save", and open the MIDI file with a program like QuickTime Player 7. - N. J. A. Sloane, Aug 08 2017
This sequence cycles clockwise around the OEIS logo. - Ryan Brooks, May 09 2020

			Consider n=6. We have a(5)=7 and try to subtract 6. The result, 1, is certainly positive, but we cannot use it because 1 is already in the sequence. So we must add 6 instead, getting a(6) = 7 + 6 = 13.

Alex Bellos and Edmund Harriss, Visions of the Universe (2016), Unnumbered pages. Includes Harriss's illustration of the first 65 steps drawn as a spiral.
Benjamin Chaffin, N. J. A. Sloane, and Allan Wilks, On sequences of Recaman type, paper in preparation, 2006.
Bernardo Recamán Santos, letter to N. J. A. Sloane, Jan 29 1991
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, The first 100000 terms
Farid Aliniaeifard and Shu Xiao Li, Peak algebra in noncommuting variables, arXiv:2506.12868 [math.CO], 2025. See p. 45.
Alex Bellos and Brady Haran, The Slightly Spooky Recamán Sequence, Numberphile video, 2018.
Alex Bellos and Brady Haran, Edmund Harriss's illustration of first 62 steps drawn as a spiral, snapshot from Numberphile video "The Slightly Spooky Recamán Sequence" (2018) at 2:37 minutes. [Included with permission of the authors] See also the Harriss link below.
Harlan Brothers, Recamán's Blues (a sonification Recamán's sequence), Animation, Jun 8, 2024.
Benjamin Chaffin, Log-log plot of first 10^230 terms
Benjamin Chaffin, Status Report, Jan 25 2018.
Fabio Deelan Cunden, Marilena Ligabò, and Tommaso Monni, Random matrices associated to Young diagrams, arXiv:2301.13555 [math.PR], 2023. See p. 7.
Michael De Vlieger, video of first 1200 steps of the Recamán sequence, with audio accompaniment generated by aspects of the sequence. Oct 12, 2019.
GBnums, A nice OEIS sequence
Gordon Hamilton, Wrecker Ball Sequences, Video, 2013
Edmund Harriss, The first 65 steps drawn as a spiral
Nick Hobson, Python program for this sequence
Bradley Klee, Code for pdf spiral drawing (golang)
Bradley Klee, The first 65 steps drawn as a spiral (color)
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
C. L. Mallows, Plot (jpeg) of first 10000 terms
C. L. Mallows, Plot (postscript) of first 10000 terms
Joseph Samuel Myers, Richard Schroeppel, Scott R. Shannon, N. J. A. Sloane, and Paul Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
Tilman Piesk, First 172 items in a coordinate system [This is a graph of the start of A005132 rotated clockwise by 90 degs. - _N. J. A. Sloane_, Mar 23 2012]
Omar E. Pol, Illustration of initial terms of A001057, A005132, A000217, 2012.
Omar E. Pol, Illustration of initial terms, 2012.
Omar E. Pol, Lateral view of a 3D-model whose front view is formed by spirals, 2022 (using Plot 2 A005132 vs A000004)
Bernardo Recamán Santos and N. J. A. Sloane, Correspondence, 1991.
Scott R. Shannon, Illustration of the first 2000 terms plotted as steps on a 2D square (Ulam) spiral. The colors are graduated across the spectrum from red to violet to show the relative step order.
Muhammad Khubab Siddique, Sequence and Series-I, Unit 8, Mathematics-II, Dept. of Sci. Ed., Allama Iqbal Open Univ. (Islamabad, Pakistan, 2020), 281-313.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, pp. 10, 12-13.
Alex van den Brandhof, Een bizarre rij, Pythagoras, 55ste Jaargang, Nummer 2, Nov 2015 (Article in Dutch about this sequence, see page 19 and back cover).
Index entries for sequences related to Recamán's sequence

Cf. A057165 (addition steps), A057166 (subtraction steps), A057167 (steps to hit n), A008336, A046901 (simplified version), A064227 (records for reaching n), A064228 (value of n that take a record number of steps to reach), A064284 (no. of times n appears), A064290 (heights of terms), A064291 (record highs), A119632 (condensed version), A063733, A079053, A064288, A064289, A064387, A064388, A064389, A228474 (bidirectional version).
A065056 gives partial sums, A160356 gives first differences.
A row of A066201.
Cf. A171884 (injective variant).
See A324784, A324785, A324786 for the "low points".
Cf. A330791, A331659, A331670.

import Data.Set (Set, singleton, notMember, insert)
a005132 n = a005132_list !! n
a005132_list = 0 : recaman (singleton 0) 1 0 where
   recaman :: Set Integer -> Integer -> Integer -> [Integer]
   recaman s n x = if x > n && (x - n) `notMember` s
                      then (x-n) : recaman (insert (x-n) s) (n+1) (x-n)
                      else (x+n) : recaman (insert (x+n) s) (n+1) (x+n)
-- Reinhard Zumkeller, Mar 14 2011

function a=A005132(m)
% m=max number of terms in a(n). Offset n:0
a=zeros(1,m);
for n=2:m
    B=a(n-1)-(n-1);
    C=0.^( abs(B+1) + (B+1) );
    D=ismember(B,a(1:n-1));
    a(n)=a(n-1)+ (n-1) * (-1)^(C + D -1);
end
% Adriano Caroli, Dec 26 2010

h := array(1..100000); maxt := 100000; a := [1]; ad := [1]; su := []; h[1] := 1; for nx from 2 to 500 do t1 := a[nx-1]-nx; if t1>0 and h[t1] <> 1 then su := [op(su), nx]; else t1 := a[nx-1]+nx; ad := [op(ad), nx]; fi; a := [op(a),t1]; if t1 <= maxt then h[t1] := 1; fi; od: # a is A005132, ad is A057165, su is A057166
A005132 := proc(n)
    option remember; local a, found, j;
    if n = 0 then return 0 fi;
    a := procname(n-1) - n ;
    if a <= 0 then return a+2*n fi;
    found := false;
    for j from 0 to n-1 while not found do
        found := procname(j) = a;
    od;
    if found then a+2*n else a fi;
end:
seq(A005132(n), n=0..70); # R. J. Mathar, Apr 01 2012 (reformatted by Peter Luschny, Jan 06 2019)

a = {1}; Do[ If[ a[ [ -1 ] ] - n > 0 && Position[ a, a[ [ -1 ] ] - n ] == {}, a = Append[ a, a[ [ -1 ] ] - n ], a = Append[ a, a[ [ -1 ] ] + n ] ], {n, 2, 70} ]; a
(* Second program: *)
f[s_List] := Block[{a = s[[ -1]], len = Length@s}, Append[s, If[a > len && !MemberQ[s, a - len], a - len, a + len]]]; Nest[f, {0}, 70] (* Robert G. Wilson v, May 01 2009 *)
RecamanSeq[i_Integer] := Fold[With[{lst=Last@#, len=Length@#}, Append[#, If[lst > len && !MemberQ[#, lst - len], lst - len, lst + len]]] &, {0}, Range@i]; RecamanSeq[10^5] (* Mikk Heidemaa, Nov 02 2024 *)

a(n)=if(n<2,1,if(abs(sign(a(n-1)-n)-1)+setsearch(Set(vector(n-1,i,a(i))),a(n-1)-n),a(n-1)+n,a(n-1)-n))  \\ Benoit Cloitre

A005132(N=1000,show=0)={ my(s,t); for(n=1,N, s=bitor(s,1<M. F. Hasler, May 11 2008, updated M. F. Hasler, Nov 03 2014

l=[0]
for n in range(1, 101):
    x=l[n - 1] - n
    if x>0 and not x in l: l+=[x, ]
    else: l+=[l[n - 1] + n]
print(l) # Indranil Ghosh, Jun 01 2017

def recaman(n):
  seq = []
  for i in range(n):
    if(i == 0): x = 0
    else: x = seq[i-1]-i
    if(x>=0 and x not in seq): seq+=[x]
    else: seq+=[seq[i-1]+i]
  return seq
print(recaman(1000)) # Ely Golden, Jun 14 2018

from itertools import count, islice
def A005132_gen(): # generator of terms
    a, aset = 0, set()
    for n in count(1):
        yield a
        aset.add(a)
        a = b if (b:=a-n)>=0 and b not in aset else a+n
A005132_list = list(islice(A005132_gen(),30)) # Chai Wah Wu, Sep 15 2022

a(k) = A000217(k) - 2*Sum_{i=1..n} A057166(i), for A057166(n) <= k < A057166(n+1). - Christopher Hohl, Jan 27 2019

Allan Wilks, Nov 06 2001, computed 10^15 terms of this sequence. At this point all the numbers below 852655 had appeared, but 852655 = 5*31*5501 was missing.
After 10^25 terms of A005132 the smallest missing number is still 852655. - Benjamin Chaffin, Jun 13 2006
Even after 7.78*10^37 terms, the smallest missing number is still 852655. - Benjamin Chaffin, Mar 28 2008
Even after 4.28*10^73 terms, the smallest missing number is still 852655. - Benjamin Chaffin, Mar 22 2010
Even after 10^230 terms, the smallest missing number is still 852655. - Benjamin Chaffin, 2018
Changed "positive" in definition to "nonnegative". - N. J. A. Sloane, May 04 2020

A135287 a(0)=1; for n > 0, a(n) = a(n-1)+n if a(n-1) is odd, else a(n) = a(n-1)/2.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 14, 7, 16, 8, 4, 2, 1, 15, 30, 15, 32, 16, 8, 4, 2, 1, 24, 12, 6, 3, 30, 15, 44, 22, 11, 43, 76, 38, 19, 55, 92, 46, 23, 63, 104, 52, 26, 13, 58, 29, 76, 38, 19, 69, 120, 60, 30, 15, 70, 35, 92, 46, 23, 83

Offset: 0

Ctibor O. Zizka, Dec 03 2007, Dec 05 2007

Let a(0), C1, C2, C be integers. Consider the sequence a(n) = a(n-1) + C1*n + C2 if a(n-1) is not divisible by C or a(n) = a(n-1)/C otherwise.
For a fixed C1, C2, C this sequence shows chaotic behavior for some a(0) and a highly regular behavior for other a(0).
The parameter C1 tells how many regular subclasses are there.
The sequence grows roughly as a(n) ~ n*const.
Here C = 2. Other sequences showing very interesting behavior have C = power of 2.
Example: C1=3, C2=10, C=3. Thus a(n)= a(n-1)+3*n+10 if a(n-1) is not divisible by 3, or a(n)= a(n-1)/3 otherwise. There are 2 classes:
a regular class with 3 subclasses (C1=3) for initial values
{a(0)=3,38,79,...}
{a(0)=1,8,12,42,47,49,63,77,88,...}
{a(0)=2,43,45,...}
and a "chaotic" class for other initial values a(0).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A008336, A005132, A135294.

Cf. A090895.

a135287 n = a135287_list !! n
a135287_list = 1 : f 1 1 where
   f x y = z : f (x + 1) z where
        z = if m == 0 then y' else x + y; (y',m) = divMod y 2
-- Reinhard Zumkeller, Mar 02 2012

A135287 := proc(n) option remember ; if n = 0 then 1 ; elif A135287(n-1) mod 2 = 0 then A135287(n-1)/2 ; else n+A135287(n-1) ; fi ; end: seq(A135287(n),n=0..60) ; # R. J. Mathar, Dec 12 2007

nxt[{n_,a_}]:={n+1,If[OddQ[a],a+n+1,a/2]}; NestList[nxt,{0,1},60][[;;,2]] (* Harvey P. Dale, Mar 02 2023 *)

More terms from R. J. Mathar, Dec 12 2007

Offset fixed by Reinhard Zumkeller, Mar 02 2012

cp's OEIS Frontend

A337486 Numbers k such that b(k+1) divides b(k), where b() is Recamán's multiplicative sequence A008336.

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A370968 A008336 sorted and duplicates removed.

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A371908 a(n) = 2-adic valuation of A008336(2*n).

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A372095 a(n) = 3-adic valuation of A008336(3*n).

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A372840 a(n) = 5-adic valuation of A008336(5*n).

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A195504 Product of numbers up to n-1 used as divisors in A008336(n), n >= 2; a(1) = 1.

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Maple

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A369407 A variant of A008336 based on polynomials over GF(2) (see Comments for precise definition).

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PARI

A370971 A008336(n) is divisible by the product of the primes p such that n/2 <= p < n; a(n) is the quotient.

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A005132 Recamán's sequence (or Recaman's sequence): a(0) = 0; for n > 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.

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