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

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

Author

Franklin T. Adams-Watters, Sep 30 2009

Keywords

• nonn

Comments

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)

Links

• 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]

Crossrefs

Cf. A007947.

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

Programs

Mathematica

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

PARI

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]))

Python

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