A091508 -id:A091508 - cp's OEIS frontend

A133058 a(0) = a(1) = 1; for n > 1, a(n) = a(n-1) + n + 1 if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1),n).

1, 1, 4, 8, 2, 8, 4, 12, 3, 1, 12, 24, 2, 16, 8, 24, 3, 21, 7, 27, 48, 16, 8, 32, 4, 30, 15, 5, 34, 64, 32, 64, 2, 36, 18, 54, 3, 41, 80, 120, 3, 45, 15, 59, 104, 150, 75, 123, 41, 91, 142, 194, 97, 151, 206, 262, 131, 189, 248, 308, 77, 139, 202, 266, 133, 199, 266, 334, 167

Offset: 0

Ctibor O. Zizka, Dec 16 2007

Remarkably, at n = 638 the sequence settles down and becomes quasi-periodic. - N. J. A. Sloane, Feb 22 2015. (Whenever I look at this sequence I am reminded of the great "Fly straight, dammit" scene in the movie "Avatar". - N. J. A. Sloane, Aug 06 2019)
For every choice of initial term a(0) there exist integers r and t >= 0 such that a(2*r+4*t+0) = 1, a(2*r+4*t+1) = 2*r+4*t+3, a(2*r+4*t+2) = 2*(2*r+4*t+3), a(2*r+4*t+3) = 2. - Ctibor O. Zizka, Dec 26 2007
See also the variants A255051 (which starts immediately with the same (1, x, 2x, 2) loop that the present sequence enters at n >= 641) and A255140 (which enters a different loop at n = 82). - M. F. Hasler, Feb 15 2015
With the recurrence used here (but with different starting values), if at some point we find a(2k) = 1, then from that point on the sequence looks like (1, x, 2x, 2), (1, x+4, 2(x+4), 2), (1, x+8, 2(x+8), 2), (1, x+12, 2(x+12), 2), ... where x = 2k+3. This is just a restatement of Zizka's comment above (although I have not seen a proof that this must always happen). - N. J. A. Sloane, Feb 22 2015
It is conjectured that quasi-periodic sequences exist only for R = 0, 1, 2 or 3 in a(n) = a(n-1) + n + R and that for R >= 4 the recurrence is not quasi-periodic. For R = 0, 1, 2 all starting values give a quasi-periodic sequence. The respective loop is (1, x) for R = 0, (1, x, 2x, 2) for R = 1 (this sequence), (1, x, 2x, x) or (2x, x) for R = 2. For R = 3 only some starting values converge to a 6-loop (4x+2, 2x+1, 3x+6, x+2, 2x+9, 3x+17). - Ctibor O. Zizka, Oct 27 2015

N. J. A. Sloane, Table of n, a(n) for n = 0..10000, May 26 2016 [First 1000 terms from Harvey P. Dale]
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
N. J. A. Sloane, Graph of the first 1000 terms, showing the transition from chaos to order more dramatically.
N. J. A. Sloane and Brady Haran, Amazing Graphs, Numberphile video (2019).
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A091508, A133579, A133580; A255051, A255140, A262922.

Quadrisections: A326991, A326992, A326993, A326994.

a:=[1]; for n in [2..70] do if Gcd(a[n-1],n) eq 1 then Append(~a, a[n-1] + n + 1); else Append(~a, a[n-1] div Gcd(a[n-1],n)); end if; end for; [1] cat a; // Marius A. Burtea, Jan 12 2020

A[0]:= 1: A[1]:= 1:
for n from 2 to 1000 do
  g:= igcd(A[n-1],n);
  A[n]:= A[n-1]/g + `if`(g=1, n+1, 0);
od:
seq(A[i],i=0..1000); # Robert Israel, Nov 06 2015

nxt[{n_,a_}]:={n+1,If[CoprimeQ[a,n+1],a+n+2,a/GCD[a,n+1]]}; Join[{1}, Transpose[ NestList[nxt,{1,1},70]][[2]]] (* Harvey P. Dale, Feb 14 2015 *)

(A133058_upto(N)=vector(N, n, if(gcd(N,n-1)>1 || n<3, N\=gcd(N,n-1), N+=n)))(100) \\ M. F. Hasler, Feb 15 2015, simplified Jan 11 2020

from itertools import count, islice
from math import gcd
def A133058_gen(): # generator of terms
    a = 1
    yield from (1,1)
    for n in count(2):
        yield (a:=a+n+1 if (b:=gcd(a,n)) == 1 else a//b)
A133058_list = list(islice(A133058_gen(),30)) # Chai Wah Wu, Mar 18 2023

More terms from Ctibor O. Zizka, Dec 26 2007

Offset and definition corrected by N. J. A. Sloane, Feb 13 2015

A381466 a(0) = 4; for n > 0, a(n) = a(n-1) + n if G = 1 or a(n) = n/G if G > 1, where G = gcd(a(n-1), n).

Original entry on oeis.org

4, 5, 7, 10, 2, 7, 13, 20, 2, 11, 21, 32, 3, 16, 7, 22, 8, 25, 43, 62, 10, 31, 53, 76, 6, 31, 57, 9, 37, 66, 5, 36, 8, 41, 75, 7, 43, 80, 19, 58, 20, 61, 103, 146, 22, 67, 113, 160, 3, 52, 25, 76, 13, 66, 9, 64, 7, 64, 29, 88, 15, 76, 31, 94, 32, 97, 163, 230, 34, 103, 173, 244

Offset: 0

Sam Chapman, Feb 24 2025

nonn

If a(n) < n for some n, then a(n+1) > n+1.
If a(n) > n, a(n+1) > n+1, and a(n+2) > n+2 for some n, then a(n+3) < n+3.
1 < a(n) for all n.
sqrt(n/6) < a(n) <= 7n/2 - 9/2 for all n.
a(p)>p for all primes p.
If one were to use the same rule to generate this sequence with any other initial value that is congruent to 4 or 8 (mod 12), that sequence would agree with this one for all n>3.
If one were to use the same rule to generate this sequence with an initial term that is not congruent to 4 or 8 (mod 12), then it would output the number 1 before the 5th term. When a sequence follows a(n)’s rules and outputs the number 1 at some index k, one gets the following quasi-periodic behavior: 1, k+2, 1, k+4, 1, k+6, etc., and are as such “boring” sequences.

			a(12) = 3 and gcd(3, 13) = 1, so a(13) = 3 + 13 = 16.  gcd(16, 14) = 2, so a(14) = 14/2 = 7.

Sam Chapman, Table of n, a(n) for n = 0..10000

Similar to A133058, A091508.

s={4};Do[G=GCD[s[[-1]],n];AppendTo[s,If[G==1,s[[-1]]+n,n/G]],{n,71}];s (* James C. McMahon, Mar 02 2025 *)

lista(nn) = my(v = vector(nn)); v[1] = 4; for (n=2, nn, my(g=gcd(v[n-1], n-1)); if (g==1, v[n] = v[n-1] + n-1, v[n] = (n-1)/g);); v; \\ Michel Marcus, Feb 26 2025

A262922 a(1)=1; for n>1, a(n) = a(n-1) + n + 2 if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1),n).

Original entry on oeis.org

1, 5, 10, 5, 1, 9, 18, 9, 1, 13, 26, 13, 1, 17, 34, 17, 1, 21, 42, 21, 1, 25, 50, 25, 1, 29, 58, 29, 1, 33, 66, 33, 1, 37, 74, 37, 1, 41, 82, 41, 1, 45, 90, 45, 1, 49, 98, 49, 1, 53, 106, 53, 1, 57, 114, 57, 1, 61, 122, 61, 1, 65, 130, 65, 1, 69, 138, 69, 1, 73, 146, 73, 1, 77, 154, 77, 1, 81, 162

Offset: 1

Ctibor O. Zizka, Oct 04 2015

nonn

This recurrence is quasi-periodic.
For some choice of starting value a(1) there exists an integer t>=1 such that a(4*t-3)=1, a(4*t-2)=4*t+1, a(4*t-1)=2*(4*t+1), a(4*t)=4*t+1. The loop is (1,x,2x,x).
For some choice of starting value a(1) there exists an integer t>=1 such that a(2*t)=2*t-1 and a(2*t-1)=2*(2*t-1). The loop is (x,2x). See also A133058.
Quasi-periodic sequences exist only for R=0,1,2 or 3 in a(n) = a(n-1) + n + R. For R=0,1,2 all starting values give a quasi-periodic sequence. The respective loop is (1,x) for R=0, (1,x,2x,2) for R=1, (1,x,2x,x) or (x,2x) for R=2. For R=3 only some starting values converge to a 6-loop (4x+2,2x+1,3x+6,x+2,2x+9,3x+17). Conjecture: For R>=4 the recurrence is not quasi-periodic.

Benoit Cloitre, 10 conjectures in additive number theory, arXiv:1101.4274 [math.NT], 2011.
Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

Cf. A091508, A106108, A133058, A133579, A133580, A255051, A255140.

a[1] = 1; a[n_] := a[n] = If[CoprimeQ[a[n - 1], n], a[n - 1] + n + 2, a[n - 1]/GCD[a[n - 1], n]]; Array[a, {79}] (* Michael De Vlieger, Oct 05 2015 *)

A=vector(1000, i, 1); for(n=2, #A, A[n]=if(gcd(A[n-1], n)>1, A[n-1]/gcd(A[n-1], n), A[n-1]+n+2))

Maple suggests the rational o.g.f. (-x^6 - x^5 - x^3 + 6x^2 + 4x + 1)/((x + 1)(x - 1)^2(x^2 + 1)^2), which should be easy to check. - Pater Bala, Oct 04 2015

A361672 a(1) = a(2) = 1; for n > 2, a(n) = a(n-2) + a(n-1) + n if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1), n).

Original entry on oeis.org

1, 1, 5, 10, 2, 1, 10, 5, 24, 12, 47, 71, 131, 216, 72, 9, 98, 49, 166, 83, 270, 135, 428, 107, 560, 280, 867, 1175, 2071, 3276, 5378, 2689, 8100, 4050, 810, 45, 892, 446, 1377, 1863, 3281, 5186, 8510, 4255, 851, 37, 935, 1020, 2004, 1002, 334, 167, 554, 277, 886

Offset: 1

Hubert W. Westwood, Mar 20 2023

Cf. A133058, A091508, A133579, A133580, A255051, A255140.

a:=[1, 1]; for n in [3..50] do if Gcd(a[n-1], n) eq 1 then Append(~a, a[n-2] + a[n-1] + n); else Append(~a, a[n-1] div Gcd(a[n-1], n)); end if; end for; [] cat a;

a[1]=a[2]=1; a[n_]:=a[n]=If[GCD[a[n-1],n]==1,a[n-2]+a[n-1]+n,a[n-1]/GCD[a[n-1],n]]; Array[a,55] (* Stefano Spezia, Mar 20 2023 *)

cp's OEIS Frontend

A133058 a(0) = a(1) = 1; for n > 1, a(n) = a(n-1) + n + 1 if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1),n).

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A381466 a(0) = 4; for n > 0, a(n) = a(n-1) + n if G = 1 or a(n) = n/G if G > 1, where G = gcd(a(n-1), n).

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A262922 a(1)=1; for n>1, a(n) = a(n-1) + n + 2 if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1),n).

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A361672 a(1) = a(2) = 1; for n > 2, a(n) = a(n-2) + a(n-1) + n if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1), n).

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