login
Numbers k that divide A000201(k)^m for some integer m > 0, where A000201 is the lower Wythoff sequence.
3

%I #16 Aug 11 2022 03:20:00

%S 1,4,8,25,50,108,169,243,256,338,486,512,729,768,972,1024,1156,1215,

%T 2312,3375,5000,7921,8192,8748,10000,12800,15000,15842,20000,25000,

%U 50176,54289,85184,88209,100352,104976,108578,131072,176418,177147

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

%C 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).

%C Many of these terms are powers of Fibonacci numbers.

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

%e For n=8, A000201(8)=12. Since 8 divides 12^2, 8 is in this sequence.

%e For n=9, A000201(9)=14. Since 9 cannot divide 14^m for any m, 9 is not in this sequence.

%o (Python)

%o from math import isqrt, prod

%o from itertools import count, islice

%o from sympy import primefactors

%o def A185615_gen(startvalue=1): # generator of terms >= startvalue

%o return filter(lambda n: not (n+isqrt(5*n**2)>>1)%prod(primefactors(n)),count(max(startvalue,1)))

%o A185615_list = list(islice(A185615_gen(),30)) # _Chai Wah Wu_, Aug 10 2022

%Y Cf. A000201, A185616, A185617.

%K nonn

%O 1,2

%A _Paul D. Hanna_ and _Sean A. Irvine_, Jan 31 2011