Enumeration of a certain sequence IV

Question

Let $F_n$ denote the $n$-th Fibonacci number. I am interested in the sequence $$a(k, n)=\left | \left \{0 \leq m \leq n: \frac{F_m}{k} \; \text{is a perfect square} \right \} \right |,$$ where $|\cdot|$ denotes the cardinality of a set.

For example, $a(1, 100) = 4$ since there are only four perfect squares among the first $n=100$ Fibonnaci numbers: $F_0=0$, $F_1=1$, $F_2=1$ and $F_{12}=144$.

How can I write a code to calculate $a(k, n)$?

@cvgmt, I think this is a common synonym for a square number. — Domen, Jun 24 '23 at 11:45
@Domen I know Perfect triangle numbers etc. but I do not know Perfect square. — cvgmt, Jun 25 '23 at 00:14

Nasser · Accepted Answer · 2023-06-24T10:45:51.867

4

You did not give an example. Is this what you want?

Using Michael E2 code to check for perfect square from Fastest square number test

ClearAll[a,sQ];
sQ[n_]:=FractionalPart@Sqrt[n+0``1]==0;
a[k_Integer,n_Integer?Positive]:=Table[f=Fibonacci[m]/k; If[sQ[f],m,Nothing],{m,0,n}];
a[1,100]

Mathematica graphics

Length[%]
(*4*)

edited Jun 24 '23 at 10:45

answered Jun 24 '23 at 09:51

Nasser

143,286
11
154
359

I am sorry for the misunderstanding. The horizontal brackets denote cardinality. – user227351 Jun 24 '23 at 10:17
$a(n,k)$ is essentially the number of integers $m$ less than $n$ such that $F_m/k$ is a perfect square. – user227351 Jun 24 '23 at 10:17
@user227351 Ok, if all what you want is the number of integers $m$ that satisfy this, then simply it will be the length of the result. No? Will update. – Nasser Jun 24 '23 at 10:23
I am not that competent, but SquareFreeQ ask "Is $n$ square free?" not "Is $n$ not a square?". – user227351 Jun 24 '23 at 10:25
@user227351 let me check again. Will delete this for now meanwhile until I am sure I understand better the question. – Nasser Jun 24 '23 at 10:26
1

@user227351 could you check if this is now what you meant? Found a test for perfect square. So in this example, the length is 3. – Nasser Jun 24 '23 at 10:41
The code is correct. Just the output should be the cardinality. – user227351 Jun 24 '23 at 10:45
I understand now, thank you very much! – user227351 Jun 24 '23 at 10:49

Enumeration of a certain sequence IV

1 Answers1