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Math Help - Finding the number of numbers with odd divisors

  1. #1
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    Finding the number of numbers with odd divisors

    hi, if we have two natural numbers say a and b then how can one find the number of numbers between them which have exactly k number of divisors?
    Here k is an odd number.
    I know that we need to find the perfect squares between these limits but how to get those who have exactly k divisors?
    E.g
    if a =2 , b =49 and k = 3 then the ans = 3
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  2. #2
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by pranay View Post
    hi, if we have two natural numbers say a and b then how can one find the number of numbers between them which have exactly k number of divisors?
    Here k is an odd number.
    I know that we need to find the perfect squares between these limits but how to get those who have exactly k divisors?
    E.g
    if a =2 , b =49 and k = 3 then the ans = 3
    for the particular example given it can be solved in the following way.

    let a number n=$p_1^$k_1$p_2^$k_2.... $p_n^$k_n where $p_i's are primes.
    then the number of divisors of n is ($k_1+1)($k_2+1)... ($k_n+1)
    now here ($k_1+1)($k_2+1)... ($k_n+1)=3
    so one of $k_i+1 is 3. this also means that only one of the $k_i is 2 and all others are 0.
    so n= p^2 for some prime p; 2<n<49.
    the only p's which qualify for the above condition are 2,3 and 5.
    so the there numbers are 4,9 and 25.
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