# P – smooth numbers in given ranges

Given multiple ranges [L, R] and a prime number p, we need to find all P-smooth numbers in given individual ranges.

What is P – smooth number?
An integer is P – smooth number if the largest Prime factor of that number <= p. 1 is considered (by OEIS) as P – smooth number for any possible value of P because it does not have any prime factor.

Examples:

```Input : p = 7
ranges[] = {[1, 17],  [10, 25]}

Output :
For first range : 1 2 3 4 5 6 7 8 9 12 14 15 16
For second range : 15 16 18 20 21 24 25
Explanation : Largest prime factors of numbers
printed above are less than or equal to 7.
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Suppose, we are checking 7 – smooth numbers.
1. Consider an integer 56. Here, 56 = 2 * 2 * 2 * 7.
So, 56 has two prime factors (2 and 7) which are <=7. So, 56 is 7-smooth number.
2. Consider another integer 66. Here, 66 = 2 * 3 * 11.
66 has three prime factors (2, 3 and 11). Where 11>7. So 66 is not 7-smooth number.

Brute – Force Approach: Let P and range [L, R] is given. Here L <= R. Create a loop and check for all numbers in inclusive range [L : R]. If that number has largest prime factor <= p. Then print that number (i.e. P-smooth number). Calculate its Largest Prime Factor / Divisor, using maxPrimeDivisor(n) function.

Efficient Approach: The idea is to pre-compute p-smooth numbers for maximum value of all ranges. Once we have pre-computed, we can quickly print for all ranges one by one.

 `# Python program to display p-smooth  ` `# number in given range. ` `# P-smooth numbers' array ` `p_smooth ``=` `[``1``]  ` ` `  `def` `maxPrimeDivisor(n): ` `     `  `    ``# Returns Maximum Prime  ` `    ``# Divisor of n ` `    ``MPD ``=` `-``1` `     `  `    ``if` `n ``=``=` `1` `:  ` `        ``return` `1` `     `  `    ``while` `n ``%` `2` `=``=` `0``: ` `        ``MPD ``=` `2` `        ``n ``=` `n ``/``/` `2` `     `  `    ``# math.sqrt(n) + 1 ` `    ``size ``=` `int``(n ``*``*` `0.5``) ``+` `1` `    ``for` `odd ``in` `range``( ``3``, size, ``2` `): ` `        ``while` `n ``%` `odd ``=``=` `0``: ` `             `  `            ``# Make sure no multiples  ` `            ``# of prime, enters here ` `            ``MPD ``=` `odd ` `            ``n ``=` `n ``/``/` `odd ` `     `  `    ``# When n is prime itself ` `    ``MPD ``=` `max` `(n, MPD)  ` `     `  `    ``return` `MPD  ` ` `  ` `  `def` `generate_p_smooth(p, MAX_LIMIT):     ` `     `  `    ``# generates p-smooth numbers. ` `    ``global` `p_smooth ` `     `  `    ``for` `i ``in` `range``(``2``, MAX_LIMIT ``+` `1``): ` `        ``if` `maxPrimeDivisor(i) <``=` `p: ` `             `  `            ``# Satisfies the condition  ` `            ``# of p-smooth number ` `            ``p_smooth.append(i) ` ` `  ` `  `def` `find_p_smooth(L, R): ` `     `  `    ``# finds p-smooth number in the ` `    ``# given [L:R] range. ` `    ``global` `p_smooth ` `    ``if` `L <``=` `p_smooth[``-``1``]: ` `         `  `        ``# If user input exceeds MAX_LIMIT ` `        ``# range, no checking ` `        ``for` `w ``in` `p_smooth : ` `            ``if` `w > R : ``break` `            ``if` `w >``=` `L ``and` `w <``=` `R : ` `                 `  `                ``# Print P-smooth numbers  ` `                ``# within range : L to R. ` `                ``print``(w, end ``=``" "``) ` `                 `  `        ``print``() ` `         `  `# p_smooth number : p = 7 ` `# L <= R ` `p ``=` `7` `L, R ``=` `1``, ``100` ` `  `# Maximum possible value of R ` `MAX_LIMIT ``=` `1000` ` `  `# generate the p-smooth numbers ` `generate_p_smooth(p, MAX_LIMIT)  ` ` `  `# Find an print the p-smooth numbers ` `find_p_smooth(L, R)  `

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