The abundancy index of a number is the ratio of the sum of its factors to the number itself.
For example a(12)=σ(n)/n=(1+2+3+4+6+12)/12 = 28/12 = 7/3.
Two numbers n and m are said to be friends if a(n)=a(m). Numbers that have at least one friend are called friendly numbers. Numbers with no friends are called solitary.
Task 1: Find at least one friend for the number 12.
Task 2: Prove there is a set of at least 50 mutual friends.
Task 3: Prove if the fraction σ(n)/n cannot be reduced then n is solitary.
Task 4: a(18)=39/18 which does reduce to 13/6. Prove 18 is solitary nevertheless.
Quite a few numbers have unknown status. The smallest of which is 10. I won't ask you to prove the status of any of these.
clearvars,clc
abundancy=cell(100000,10000);
for n=1:10000
nu=sum(divisors(n));
g=gcd(nu,n);
nu=nu/g; de=n/g;
abundancy{nu,de}{end+1}=n;
end
creates a cell array whose subscripts are the numerator and denominator of the reduced fractions of the abundancy indices of the numbers up to 10,000.
To find friend to 12:
>> abundancy{7,3}
ans =
1×2 cell array
{[12]} {[234]}
The number 234 has an abundancy of 546/234 = 7/3 as the divisors of 234 are:
1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234
To show 18 is solitary:
>> abundancy{13,6}
ans =
1×1 cell array
{[18]}
But this shows only that no number that's 10,000 or less is a friend; it could have friends in higher places.
Finding 50 or more mutual friends:
The most I could find:
8 3 4
>> abundancy{8,3}
ans =
1×4 cell array
{[84]} {[270]} {[1488]} {[1638]}
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Posted by Charlie
on 2023-06-02 12:13:13 |
#1 and a start at #4 and failed attempt at 2
