All infinite sequences, like

n_i, i=1...infinity, i'th number in a sequence

can be given algorithms

A_j, j=1...infinity, j'th algorithm in a sequence of algorithms

that deliver the numbers in the sequence n_i.

Let's say this sequence n_i describes one subject in a space, where different numbers are describing different scales in that space (like digits in decimal notation give 'different scale' properties for a physical quantity).

In order to seek for example fractal laws of subjects behaviour in different scales we would like to find a certain known sequence of numbers

m_k, k=1...K

(where K is a positive integer) from this infinite sequence of numbers. That is, we are looking for a known sequence of numbers from an infinite sequence of numbers given by algorithms.

My question is: can we easily get an algorithm, that would tell us exactly ALL locations in the infinite sequence that this known sequence could be found from?

k_l, l=1...N, N is a positive integer

This information could be useful in studying the fractal structure at different scales.

A related question is that I would like to know how many times at which decimal locations my phone number could be found from the sequence of digits in pi, or any other irrational number. This would give me the possibility to tell my phone number easier to any of my friends, simply saying:

n digits from pi starting from the m:th.

Another related problem is that this sequence of numbers could instead be multidimensional multivalued sequences and we would be looking for certain known subsequences as a general question: at which locations do they exist if we know the algorithms?