A272636 -id:A272636 - cp's OEIS frontend

A272639 Numbers n such that the sequence defined by {b(1)=1, b(2)=n, b(k) = squarefree part of b(k-1)+b(k-2) for k>2} does not eventually merge with the period-6 sequence A272636.

79, 83, 92, 101, 176, 258, 425, 445, 518, 679, 733, 758, 839, 932, 1272, 1311, 1382, 1432, 1453, 1454, 1495, 1899, 1904, 1991, 2207, 2317, 2444, 2450, 2570, 2609, 2733, 2736, 2773, 2783, 2961, 3038, 3076, 3166, 3291, 3370, 3384, 3416, 3456, 3473, 3539, 3545

Offset: 1

N. J. A. Sloane, May 05 2016

nonn

The sequences b(n) for the initial terms 79, 83, 92, 101, 176, 258, 425, 445 all merge with the period-43 sequence A272637. In fact for n=83 b(n) IS A272637.
From Lars Blomberg, Jan 02 2018: (Start)
Among the first 10000 terms, two additional cycles appear:
A) with period 22: [15, 2162, 2177, 4339, 181, 1130, 1311, 2441, 938, 3379, 4317, 481, 4798, 5279, 10077, 3839, 71, 3910, 3981, 7891, 742, 8633].
B) with period 26: [23, 1801, 114, 1915, 2029, 986, 335, 1321, 46, 1367, 157, 381, 538, 919, 1457, 66, 1523, 1589, 778, 263, 1041, 326, 1367, 1693, 85, 1778].
The first occurrence of cycle A is n=28243, for cycle B n=21245.
Cycle A occurs for 17 terms, B for 24 terms and A272637 for the remaining 9959 terms. (End)

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A007913, A272636, A272637, A272638.

Terms a(9) and beyond from Lars Blomberg, Jan 02 2018

A214674 Conway's subprime Fibonacci sequence.

Original entry on oeis.org

1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, 15, 23, 19, 21, 20, 41, 61, 51, 56, 107, 163, 135, 149, 142, 97, 239, 168, 37, 41, 39, 40, 79, 17, 48, 13, 61, 37, 49, 43, 46, 89, 45, 67, 56, 41, 97, 69, 83, 76, 53, 43, 48, 13

Offset: 1

Wouter Meeussen, Jul 25 2012

Similar to the Fibonacci recursion starting with (1, 1), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 18 after 38 terms on reaching (48, 13).

Siobhan Roberts, Genius At Play: The Curious Mind of John Horton Conway, Bloomsbury, 2015, pages xx-xxi.

Peter Kagey, Table of n, a(n) for n = 1..250
Sara Barrows, Emily Noye, Sarah Uttormark, and Matthew Wright, Three's A Crowd: An Exploration of Subprime Tribonacci Sequences, College Math. J. (2023).
Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT], 2012-2014.
Tanya Khovanova, Conway’s Subprime Fibonacci Sequences, Math Blog, July 2012.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000045, A020639, A117339, A175723, A214551, A014682, etc.
Cf. A214892-A214898, A282812, A282813, A282814.
See also A165911, A272636, A255562, etc.

guyKhoSal[{a_, b_}] := Block[{c, l, r}, c = NestWhile[(p = Tr[Take[#, -2]]; If[PrimeQ[p], q = p, q = p/Part[FactorInteger[p, FactorComplete -> False], 1, 1]]; Flatten[{#, q}]) &, {a, b}, FreeQ[Partition[#1, 2, 1], Take[#2, -2]] &, 2, 1000]; l = Length[c]; r = Tr@Position[Partition[c,2,1], Take[c,-2], 1, 1]; l-r-1; c]; guyKhoSal[{1,1}]
f[s_List] := Block[{a = s[[-2]] + s[[-1]]}, If[ PrimeQ[a], Append[s, a], Append[s, a/FactorInteger[a][[1, 1]] ]]]; Nest[f, {1, 1}, 73] (* Robert G. Wilson v, Aug 09 2012 *)

fatw(n,a=[0,1],p=[])={for(i=2,n,my(f=factor(a[i]+a[i-1])~);for(k=1,#f,setsearch(p,f[1,k])&next;f[2,k]--;p=setunion(p,Set(f[1,k]));break);a=concat(a,factorback(f~)));a}
fatw(99) /* M. F. Hasler, Jul 25 2012 */

A165911 a(n) = squarefree kernel (or radical) of a(n-1) + a(n-2), with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 7, 3, 10, 13, 23, 6, 29, 35, 2, 37, 39, 38, 77, 115, 6, 11, 17, 14, 31, 15, 46, 61, 107, 42, 149, 191, 170, 19, 21, 10, 31, 41, 6, 47, 53, 10, 21, 31, 26, 57, 83, 70, 51, 11, 62, 73, 15, 22, 37, 59, 6, 65, 71, 34, 105, 139, 122, 87, 209, 74, 283, 357, 10

Offset: 0

Franklin T. Adams-Watters, Sep 30 2009

nonn

The squarefree kernel (or radical) of n is the largest squarefree divisor of n, A007947.
Through n=1688, this sequence does not loop. Does it grow indefinitely, or is it eventually periodic?
The graph suggests that the sequence had a chance to go into a cycle between terms 100 and 150, but by the time we get to 1688 terms the sequence seems to have reached escape velocity and there is no further hope of this happening. (Of course this is not a rigorous argument.) - N. J. A. Sloane, May 06 2016
If we take the squarefree part (A007913) instead of the squarefree kernel, the sequence is periodic from n=1, repeating 1,2,3,5,2,7. See A272636.
From Fred W. Helenius, May 07 2016: (Start)
There are many examples of initial conditions for this recurrence that produce sequences that cycle.
Examples that arise where a(0) and a(1) are not coprime:
  - 2, 2 (period 1)
  - 3, 3, 6, 3, 3, 6 (period 3)
  - 5, 10, 15, 5, 10, 15 (period 3)
Examples of periodic sequences starting with coprime initial values:
  - 15 146 161 307 78 385 463 106 569 (period 9)
  - 222 1589 1811 170 1981 717 2698 3415 6113 2382 8495 10877 9686 20563 10083 30646 3133 33779 4614 38393 43007 4070 47077 17049 64126 16235 26787 6146 32933 39079 36006 75085 111091 5818 5083 10901 (period 46)
  - 770 559 1329 118 1447 1565 1506 3071 4577 478 5055 5533 5294 1203 6497 (period 15)
(End)

Franklin T. Adams-Watters and N. J. A. Sloane, Table of n, a(n) for n = 0..1688, May 07 2016 [First 1000 terms from Franklin T. Adams-Watters]

Cf. A007947.

See A000045, A272636, A272637, A272638, A272639 for similar sequences. See also A214674, A214892-A214898.

nxt[{a_,b_}]:={b,Select[Divisors[a+b],SquareFreeQ][[-1]]}; NestList[nxt,{0,1},70][[All,1]] (* Harvey P. Dale, Jul 31 2018 *)

rad(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,1])
v=vector(100,n,1);for(n=3,100,v[n]=rad(v[n-1]+v[n-2]))

from operator import mul
from sympy import primefactors
from functools import reduce
def rad(n): return 1 if n<2 else reduce(mul, primefactors(n))
l=[0, 1]
for n in range(2, 101):
    l.append(rad(l[n - 1] + l[n - 2]))
print(l) # Indranil Ghosh, Jun 03 2017

A272637 a(1)=1, a(2)=82; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

Original entry on oeis.org

1, 83, 21, 26, 47, 73, 30, 103, 133, 59, 3, 62, 65, 127, 3, 130, 133, 263, 11, 274, 285, 559, 211, 770, 109, 879, 247, 1126, 1373, 51, 89, 35, 31, 66, 97, 163, 65, 57, 122, 179, 301, 30, 331, 1, 83, 21, 26, 47, 73, 30, 103, 133, 59, 3, 62, 65, 127, 3, 130, 133, 263, 11, 274, 285, 559, 211, 770, 109, 879, 247, 1126, 1373, 51, 89, 35, 31, 66, 97, 163, 65, 57, 122, 179, 301, 30, 331

Offset: 1

N. J. A. Sloane, May 05 2016

nonn

Periodic with period length 43. The period is [1, 83, 21, 26, 47, 73, 30, 103, 133, 59, 3, 62, 65, 127, 3, 130, 133, 263, 11, 274, 285, 559, 211, 770, 109, 879, 247, 1126, 1373, 51, 89, 35, 31, 66, 97, 163, 65, 57, 122, 179, 301, 30, 331].

Fred W. Helenius, Posting to Math Fun Mailing List, May 06 2016

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

Cf. A272636.

A272638 a(1)=1, a(2)=4; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

Original entry on oeis.org

1, 4, 5, 1, 6, 7, 13, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7

Offset: 1

N. J. A. Sloane, May 05 2016

nonn

Rapidly merges with A272636.

Cf. A000045, A007913.

sqrf:= n-> mul(i[1]^irem(i[2], 2), i=ifactors(n)[2]):
f:=proc(n) global sqrf; option remember;
if n = 1 then 1 elif n=2 then 4 else sqrf(f(n-1)+f(n-2)); fi; end;
[seq(f(n),n=1..140)];

cp's OEIS Frontend

A272639 Numbers n such that the sequence defined by {b(1)=1, b(2)=n, b(k) = squarefree part of b(k-1)+b(k-2) for k>2} does not eventually merge with the period-6 sequence A272636.

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A214674 Conway's subprime Fibonacci sequence.

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A165911 a(n) = squarefree kernel (or radical) of a(n-1) + a(n-2), with a(0)=0 and a(1)=1.

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A272637 a(1)=1, a(2)=82; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

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A272638 a(1)=1, a(2)=4; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

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