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 Some questions of abundance (Posted on 2015-10-28)
Under what conditions is the number 2n*p an abundant number? Where n is a positive integer and p is prime.

Recall an abundant number is one whose proper factors sum to more than the number. For example 18 is abundant because 1+2+3+6+9 > 18.

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2^n * p has 2*(n+1) proper factors when p is not equal to 2, and 2*(n+2) proper factors if p = 2.

case 1 -- p>2: The 2*(n+1) factors are 1, 2, ... , 2^n and the same numbers, except for the last, times p.  The total is (p+1)*((2^(n+1))-1) - p*2^n. This is more than p*2^n when n >= 2.

case 2 -- p=2: 2^n * p = 2^(n+1), which has proper factors 1, 2, 4, ... , 2^n, which add up to 2^(n+1) - 1, so in this case the number is never abundant.

In sum: 2^n * p is abundant unless p=2 or n = 1.

3^n * p has 2*(n+1) proper factors when p is not equal to 3, and 2*(n+2) proper factors if p = 3.

case 1 -- p<>3: The 2*(n+1) factors are 1, 3, ... , 3^n and the same numbers, except for the last, times p.  The total without the "times p" is (3^(n+1)-1)/2 as a repunit in base 3, making the total (p+1)*(3^(n+1)-1)/2 - p*3^n.

I don't think these ever become abundant.

case 2 -- p=3: 3^n * p = 3^(n+1), which has proper factors 1, 3, 9, ... , 3^n, which add up to (3^(n+1) - 1)/2, so in this case also the number is never abundant.

2*p^n , in the case where p is 2, has already been covered as p*2^n where p is 2, and found not to be abundant.

2*p^n where p is not equal to 2 has 2*(n+1) proper factors: 1, p, p^2, ... p^n and also twice each of those except for the last. The first part of the sum has the total (p^(n+1)-1)/(p-1), so the overall sum is 2*(p^(n+1)-1)/(p-1) - 2*p^n. These are also never abundant.

Of the three cases given, only 2^n * p has abundant subcases: those where p>2 and n>1.

 Posted by Charlie on 2015-10-28 10:38:43

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