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

Author

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

Keywords

• nonn

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).

Examples

			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.
		

Crossrefs

Cf. A000201, A185616, A185617.

Programs

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