Help Lakshya!

Lakshya is learning number theory. He reads about the Mobius function : \( \mu : \mathbb{N} \longrightarrow \mathbb{Z} \) defi ned by \( \mu(1) = 1\) and

\[\large{\mu \left( n \right) =-\sum _{ d|n\\ d\neq n }^{ }{ \mu \left( d \right) } }\]

for n>1. However, Lakshya doesn't like negative numbers, so he invents his own function: the Lakshyaous function \( \delta : \mathbb{N} \longrightarrow \mathbb{N}\) defined by the relations \(\delta (1)=1\) and

\[\large{\delta \left( n \right) =\sum _{ d|n\\ d\neq n }^{ }{ \delta \left( d \right) } }\]

for n > 1. So he asks his teacher Otto Bretscher to help solve the below question. Otto Bretscher gave him hint but he cannot solve. Help Lakshya determine the value of \(1000p+q\), where \(p\) and \( q\) are relatively prime positive integers satisfying:

\[\large{\frac { p }{ q } =\sum _{ k=0 }^{ \infty }{ \frac { \delta \left( { 15 }^{ k } \right) }{ { 15 }^{ k } } } }\]

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