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Find sum of natural numbers satisfying the following condition

Source:

August 11, 2019

Natural Numbersfunction

Problem Statement

A natural number

k > 1

is called good if there exist natural numbers

a_1 < a_2 < \cdots < a_k

such that

\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1

.
Let

f(n)

be the sum of the first

n

[good numbers,

n \geq

1. Find the sum of all values of

n

for which

f(n+5)/f(n)

is an integer.