A272636 a(0)=0, a(1)=1; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

0, 1, 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, 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: 0

Author

N. J. A. Sloane, May 05 2016

Keywords

• nonn

Comments

Periodic with period {1,2,3,5,2,7}.

James Propp, in a posting to the Math Fun list, asks if every sequence of positive numbers satisfying the same recurrence will eventually merge with this sequence (as A272638 does). The answer is no, Fred W. Helenius found infinitely many counterexamples, including A272637. See A272639 for other counterexamples which start 1,x.

Other counterexamples found by Helenius include [n, 2n, 3n, 5n, 2n, 7n] (period 6) where n is any squarefree positive integer coprime to 210 = 2*3*5*7.

Links

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

Crossrefs

Cf. A007913 (squarefree part of n), A000045, A272637, A272638, A272639.

See A165911 for a similar sequence.

See also A214674, A214892-A214898.

Programs

Mathematica

{0, 1}~Join~LinearRecurrence[{0, 0, 0, 0, 0, 1}, {1, 2, 3, 5, 2, 7}, 120] (* Jean-François Alcover, Nov 16 2019 *)

Python

from sympy.ntheory.factor_ import core
l=[0, 1]
for n in range(2, 101):
    l.append(core(l[n - 1] + l[n - 2]))
print(l) # Indranil Ghosh, Jun 03 2017