Related Articles

# P – smooth numbers in given ranges

• Difficulty Level : Hard
• Last Updated : 01 Mar, 2018

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.

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

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) `

My Personal Notes arrow_drop_up