A133058 -id:A133058 - cp's OEIS frontend

A326991 First quadrisection of A133058.

1, 2, 3, 2, 3, 48, 4, 34, 2, 3, 3, 104, 41, 97, 131, 77, 133, 167, 95, 80, 82, 43, 90, 47, 33, 33, 254, 8, 284, 432, 6, 47, 107, 125, 265, 341, 385, 8, 384, 55, 31, 130, 350, 56, 200, 41, 295, 217, 69, 327, 15, 103, 210, 107, 73, 17, 2, 3, 3, 552, 83, 203, 131, 221, 73, 85, 437, 574, 347, 293, 113, 481, 636

Offset: 0

Omar E. Pol, Aug 11 2019

nonn

After the initial chaos, starting with n = 160 we have the spoke 1282, 1290, 1298, 1306, ...

Cf. A008586, A133058, A326992, A326993, A326994.

a(n) = A133058(4n) = A133058(A008586(n)).

For n >= 160; a(n) = 8*n + 2.

A326992 Second quadrisection of A133058.

Original entry on oeis.org

1, 8, 1, 16, 21, 16, 30, 64, 36, 41, 45, 150, 91, 151, 189, 139, 199, 237, 169, 158, 164, 129, 180, 141, 131, 135, 360, 118, 398, 48, 128, 173, 237, 259, 403, 483, 77, 158, 128, 213, 193, 26, 520, 230, 378, 223, 59, 31, 263, 525, 5, 309, 420, 321, 291, 1, 228, 233, 237, 184, 325, 29, 381, 475, 331

Offset: 0

Omar E. Pol, Aug 11 2019

nonn

After the initial chaos, starting with n = 160 we have the spoke 2, 2, 2, 2, ...

Cf. A016813, A133058, A326991, A326993, A326994.

a(n) = A133058(4n+1) = A133058(A016813(n)).

For n >= 160; a(n) = 2.

A326993 Third quadrisection of A133058.

Original entry on oeis.org

4, 4, 12, 8, 7, 8, 15, 32, 18, 80, 15, 75, 142, 206, 248, 202, 266, 308, 244, 79, 2, 3, 2, 3, 230, 45, 180, 59, 199, 24, 64, 300, 368, 394, 542, 626, 224, 79, 64, 372, 356, 13, 52, 115, 189, 406, 246, 222, 458, 175, 208, 3, 2, 3, 510, 224, 114, 464, 79, 92, 568, 276, 632, 730, 590, 610, 37, 422, 896, 850

Offset: 0

Omar E. Pol, Aug 11 2019

nonn

After the initial chaos, starting with n = 159 we have the spoke 1, 1, 1, 1, ...

Cf. A016825, A133058, A326991, A326992, A326994.

a(n) = A133058(4n+2) = A133058(A016825(n)).

For n >= 159; a(n) = 1.

A326994 Fourth quadrisection of A133058.

Original entry on oeis.org

8, 12, 24, 24, 27, 32, 5, 64, 54, 120, 59, 123, 194, 262, 308, 266, 334, 380, 320, 1, 86, 1, 94, 99, 330, 149, 288, 171, 315, 144, 188, 428, 500, 530, 682, 770, 32, 231, 220, 124, 520, 181, 224, 23, 369, 590, 434, 414, 654, 375, 412, 1, 214, 219, 170, 448, 342, 696, 315, 332, 812, 524, 884, 146, 850, 874, 305

Offset: 0

Omar E. Pol, Aug 11 2019

nonn

After the initial chaos, starting with n = 159 we have the spoke 641, 645, 649, 653, ...

Cf. A004767, A133058, A326991, A326992, A326993.

a(n) = A133058(4n+3) = A133058(A004767(n)).

For n >= 159; a(n) = 4*n + 5.

A309627 Starting numbers k such that the trajectory of k under the map x -> A133058(x) joins A133058.

Original entry on oeis.org

6, 8, 11, 12, 19, 25, 30, 32, 45, 46, 47, 52, 53, 55, 58, 59, 60, 63, 64, 76, 95, 97, 98, 99, 101, 102, 107, 108, 114, 118, 124, 126, 132, 134, 137, 140, 144, 152, 156, 157, 159, 163, 169, 173, 177, 180, 181, 185, 187, 188, 189, 198, 199, 200, 202, 203, 206, 207, 208, 209, 210

Offset: 1

Tomas Tkac, Aug 10 2019

nonn

			Quite a few starting numbers f(0) for the recurrence defined in A133058 will produce a sequence which eventually join A133058, the smallest such number being 6. 8 joins 6 at the 10th term, 11 joins 6 at the 17th term, 12 joins the sequence at the 97th term, etc.

Tomas Tkac, Table of n, a(n) for n = 1..306 [recomputed and restored by _Georg Fischer_, Oct 14 2019]
Brady Haran and N. J. A. Sloane, Amazing Graphs, Numberphile Video.

Cf. A133058.

Entry revised by Editors of the OEIS, Oct 01 2019

A339571 A133058 with duplicates removed.

Original entry on oeis.org

1, 4, 8, 2, 12, 3, 24, 16, 21, 7, 27, 48, 32, 30, 15, 5, 34, 64, 36, 18, 54, 41, 80, 120, 45, 59, 104, 150, 75, 123, 91, 142, 194, 97, 151, 206, 262, 131, 189, 248, 308, 77, 139, 202, 266, 133, 199, 334, 167, 237, 380, 95, 169, 244, 320, 158, 79, 82, 164, 86

Offset: 1

N. J. A. Sloane, Dec 09 2020

nonn

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 [math.CO], Dec 08, 2020. See Section 2.6.

Cf. A133058.

DeleteDuplicates@ Block[{a = {1, 1}, k = 1}, Do[AppendTo[a, If[# == 1, a[[-1]] + i + 1, a[[-1]]/#]] &@ GCD[a[[-1]], i], {i, 2, 80}]; a] (* Michael De Vlieger, Dec 09 2020 *)

lista(nn) = my(v=List([1]), x=1, y); print1(1); for(n=2, nn, if(!setsearch(Set(v), x=if(1==y=gcd(x, n), x+n+1, x/y)), print1(", ", x); listput(v, x))); \\ Jinyuan Wang, Dec 12 2020

from math import gcd
from itertools import count, islice
def A339571_gen(): # generator of terms
    a, aset = 1, {1}
    yield 1
    for n in count(2):
        a = a+n+1 if (b:=gcd(a,n)) == 1 else a//b
        if a not in aset:
            aset.add(a)
            yield a
A339571_list = list(islice(A339571_gen(),30)) # Chai Wah Wu, Mar 18 2023

More terms from Jinyuan Wang, Dec 12 2020

A133579 a(0)=a(1)=1; for n > 1, a(n) = 3*a(n-1) 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, 3, 1, 3, 9, 3, 9, 27, 3, 9, 27, 9, 27, 81, 27, 81, 243, 27, 81, 243, 81, 243, 729, 243, 729, 2187, 81, 243, 729, 243, 729, 2187, 729, 2187, 6561, 729, 2187, 6561, 2187, 6561, 19683, 6561, 19683, 59049, 6561, 19683, 59049, 19683, 59049, 177147

Offset: 0

Ctibor O. Zizka, Dec 26 2007

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A133058, A133580, A253092.

f:=proc(n) option remember;
if n <= 1 then 1
elif gcd(n,f(n-1))>1 then f(n-1)/gcd(n,f(n-1))
else 3*f(n-1); fi; end;
[seq(f(n),n=0..50)];
# N. J. A. Sloane, Feb 14 2015

nxt[{n_,a_}]:={n+1,If[CoprimeQ[a,n+1],3a,a/GCD[a,n+1]]}; Join[{1}, Transpose[ NestList[nxt,{1,1},50]][[2]]] (* Harvey P. Dale, Feb 14 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]*3)) \\ M. F. Hasler, Feb 15 2015

Offset, definition, and terms corrected by N. J. A. Sloane, Feb 14 2015

A255140 a(1) = 1, a(n+1) = a(n)/gcd(a(n),n) if this gcd is > 1, else a(n+1) = a(n) + n + 2.

Original entry on oeis.org

1, 4, 2, 7, 13, 20, 10, 19, 29, 40, 4, 17, 31, 46, 23, 40, 5, 24, 4, 25, 5, 28, 14, 39, 13, 40, 20, 49, 7, 38, 19, 52, 13, 48, 24, 61, 99, 138, 69, 23, 65, 108, 18, 63, 109, 156, 78, 127, 177, 228, 114, 38, 19, 74, 37, 94, 47, 106, 53, 114, 19, 82, 41, 106

Offset: 1

M. F. Hasler, Feb 15 2015

A variant of A133058, less trivial than A255051: The sequence looks irregular up to index n = 82, where it enters a 4-periodic pattern (1, x, 2x, x), cf. formula. Sequence A255051 starts right from the beginning with the pattern (1, x, 2x, 2), whereas sequence A133058 enters such a pattern only at index n = 641.

			a(2) = a(1) + 3 = 4, a(3) = a(2)/2 = 2, a(4) = a(3) + 5 = 7, a(5) = a(4) + 6 = 13, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A133058, A255051.

a:=[1]; for n in [2..65] do if Gcd(a[n-1],n-1) gt 1 then Append(~a, a[n-1] div Gcd(a[n-1],n-1)); else Append(~a, a[n-1] +n+1); end if; end for; a; // Marius A. Burtea, Jan 11 2020

nxt[{n_,a_}]:=Module[{g=GCD[a,n]},{n+1,If[g>1,a/g,a+n+2]}]; NestList[nxt,{1,1},70][[All,2]] (* Harvey P. Dale, Oct 12 2019 *)

A255140_vec(N)=vector(N, n, if(gcd(N,n-1)>1||n==1, N/=gcd(N, n-1), N+=n+1)) \\ Original code simplified by M. F. Hasler, Jan 11 2020

A255140(n)=if(n < 82, A255140_upto(n)[n], [2*n+2,n,1,n+2][n%4+1]) \\ M. F. Hasler, Jan 17 2020

For k > 20, a(4k) = 8k + 2 = 2*a(4k +- 1), a(4k - 2) = 1; equivalently:

a(n) = 2n + 2, n, 1 or n+2 when n = 4k+r > 81 with r = 0, 1, 2 or 3, respectively.

Edited by M. F. Hasler, Jan 11 2020

A133580 a(0)=a(1)=1; for n>1, a(n) = 2*a(n-1) + 1 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, 3, 1, 3, 7, 15, 31, 63, 7, 15, 31, 63, 127, 255, 17, 35, 71, 143, 287, 575, 1151, 2303, 4607, 9215, 1843, 3687, 1229, 2459, 4919, 9839, 19679, 39359, 78719, 157439, 314879, 629759, 1259519, 2519039, 5038079, 10076159, 20152319, 40304639

Offset: 0

Ctibor O. Zizka, Dec 26 2007

nonn

The initial value a(0)=1 is somehow artificial; using a(0)=0 would yield the same subsequent terms using the recurrence formula already for n=1. - M. F. Hasler, Feb 15 2015

			Write the GCD of a(n-1) and n under a(n-1):
n = : 0 1 2 3 4 5  6  7  8 9 ...
a(n): 1 1 3 1 3 7 15 31 63 7 ...
gcd : 1 1 3 1 1 1  1  1  9 1 ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A133058, A133579.

a = {1, 1}; Do[If[GCD[n, a[[ -1]]] == 1, b = 2*a[[ -1]] + 1, b = a[[ -1]]/GCD[a[[ -1]], n]]; AppendTo[a, b], {n, 2, 50}]; a (* Stefan Steinerberger, Dec 31 2007 *)
nxt[{a_,b_}]:={a+1,If[CoprimeQ[b,a+1],2b+1,b/GCD[b,a+1]]}; Join[{1}, Transpose[ NestList[nxt,{1,1},50]][[2]]] (* Harvey P. Dale, Sep 16 2012 *)

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]*2+1)) \\ M. F. Hasler, Feb 15 2015

a=0;#A133580=vector(1000,n,a=if(gcd(a,n)>1,a/gcd(a,n),a*2+1)) \\ M. F. Hasler, Feb 15 2015

Corrected and extended by Stefan Steinerberger, Dec 31 2007
Offset changed to 0 by N. J. A. Sloane, Feb 13 2015
The Mathematica programs are correct; b-file corrected by Harvey P. Dale, Feb 14 2015

A255051 a(1)=1, a(n+1) = a(n)/gcd(a(n),n) if this GCD is > 1, else a(n+1) = a(n) + n + 1.

Original entry on oeis.org

1, 3, 6, 2, 1, 7, 14, 2, 1, 11, 22, 2, 1, 15, 30, 2, 1, 19, 38, 2, 1, 23, 46, 2, 1, 27, 54, 2, 1, 31, 62, 2, 1, 35, 70, 2, 1, 39, 78, 2, 1, 43, 86, 2, 1, 47, 94, 2, 1, 51, 102, 2, 1, 55, 110, 2, 1, 59, 118, 2, 1, 63, 126, 2, 1, 67, 134, 2, 1, 71, 142, 2, 1

Offset: 1

M. F. Hasler, Feb 15 2015

A somehow "trivial" variant of A133058 and A255140, both of which have very similar definitions, but enter 4-periodic loops only at later indices.

There could be two motivated values for an initial term: either a(0)=0 which would yield a(1)=1 and the following values via the recursion formula, or a(0)=2 according to the general formula for a(4k).

			a(2) = a(1)+2 = 3, a(3) = a(2)+3 = 6, a(4) = a(3)/3 = 2, a(5) = a(4)/2 = 1;
a(6) = a(5)+6 = 7, a(7) = a(6)+7 = 14, a(8) = a(7)/7 = 2, a(9) = a(8)/2 = 1; ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A133058, A255140.

&cat [[1, 4*n+3, 8*n+6, 2]: n in [0..20]]; // Bruno Berselli, Feb 16 2015

Table[(2 (3 + (-1)^n) - (2 - 3 n + n (-1)^n) (1 - (-1)^((n - 1) n/2)))/4, {n, 1, 80}] (* Bruno Berselli, Feb 16 2015 *)
nxt[{n_,a_}]:={n+1,If[GCD[a,n]>1,a/GCD[a,n],a+n+1]}; Transpose[ NestList[ nxt, {1,1},80]][[2]] (* or *) LinearRecurrence[{0,0,0,2,0,0,0,-1},{1,3,6,2,1,7,14,2},80] (* Harvey P. Dale, Oct 13 2015 *)

(A255051_upto(N)=vector(N, n, if(gcd(N, n-1)>1, N\=gcd(N, n-1), N+=n)))(99) \\ simplified by M. F. Hasler, Jan 11 2020

A255051(n)=if(n%4>1,if(bittest(n,0),n*2,n+1),2-bittest(n,0)) \\ M. F. Hasler, Feb 18 2015

a(4k+1) = 1, a(4k+2) = 4k+3, a(4k+3) = 2*a(4k+2) = 8k+6, a(4k) = 2.
G.f.: x*(1 + 3*x + 6*x^2 + 2*x^3 - x^4 + x^5 + 2*x^6 - 2*x^7)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2). - Bruno Berselli, Feb 16 2015
a(n) = ( 2*(3 + (-1)^n) - (2 - 3*n + n*(-1)^n)*(1 - (-1)^((n-1)*n/2)) )/4. - Bruno Berselli, Feb 16 2015

cp's OEIS Frontend

A326991 First quadrisection of A133058.

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A326992 Second quadrisection of A133058.

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A326993 Third quadrisection of A133058.

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A326994 Fourth quadrisection of A133058.

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A309627 Starting numbers k such that the trajectory of k under the map x -> A133058(x) joins A133058.

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A339571 A133058 with duplicates removed.

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A133579 a(0)=a(1)=1; for n > 1, a(n) = 3*a(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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A255140 a(1) = 1, a(n+1) = a(n)/gcd(a(n),n) if this gcd is > 1, else a(n+1) = a(n) + n + 2.

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A133580 a(0)=a(1)=1; for n>1, a(n) = 2*a(n-1) + 1 if a(n-1) and n are coprime, otherwise a(n) = a(n-1)/gcd(a(n-1),n).

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