# Bitter Numbers in the Bad Hall of Notoriety

Number Theory Level pending

The president of Sugar Land has created this iterative rule for generating sequences of whole numbers:

1) If a number is $$25$$ or less, double that number.

2) If a number is greater than $$25$$, subtract $$12$$ from it.

Let $$S$$ be the first number in a sequence created by the two rules above. $$S$$ is considered a "sweet" number if the sequence that starts with $$S$$ contains the number $$16$$. If not, then $$S$$ is considered a bitter number and banned from Sugar Land. Furthermore, if the bitter number is divisible by $$5$$, then it is placed in the Bad Numbers Hall of Notoriety in Bitter Nation. How many numbers from $$1$$ to $$1,000$$ are in the Bad Numbers Hall of Notoriety in Bitter Nation?

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