A185615 Numbers k that divide A000201(k)^m for some integer m > 0, where A000201 is the lower Wythoff sequence.

1, 4, 8, 25, 50, 108, 169, 243, 256, 338, 486, 512, 729, 768, 972, 1024, 1156, 1215, 2312, 3375, 5000, 7921, 8192, 8748, 10000, 12800, 15000, 15842, 20000, 25000, 50176, 54289, 85184, 88209, 100352, 104976, 108578, 131072, 176418, 177147

Offset

1,2

Comments

Let k = p_1^{e_1} * p_2^{e_2} * ... * p_r^{e_r}. Then k is in this sequence iff p_1*p_2*...*p_r divides A000201(k).

Many of these terms are powers of Fibonacci numbers.

Perhaps this is expected, since A000201(k) involves floor(k*phi).

Links

Table of n, a(n) for n=1..40.

Example

For n=8, A000201(8)=12. Since 8 divides 12^2, 8 is in this sequence.
For n=9, A000201(9)=14. Since 9 cannot divide 14^m for any m, 9 is not in this sequence.

Prog

(Python)
from math import isqrt, prod
from itertools import count, islice
from sympy import primefactors
def A185615_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not (n+isqrt(5*n**2)>>1)%prod(primefactors(n)), count(max(startvalue, 1)))
A185615_list = list(islice(A185615_gen(), 30)) # Chai Wah Wu, Aug 10 2022

Crossrefs

Cf. A000201, A185616, A185617.

Sequence in context: A163143 A154586 A371190 * A068367 A292548 A000964

Adjacent sequences: A185612 A185613 A185614 * A185616 A185617 A185618

Keyword

nonn

Author

Paul D. Hanna and Sean A. Irvine, Jan 31 2011

Status

approved