Duffinian Numbers

Duffinian numbers are defined as numbers that are composite and relatively prime to the sum of their divisors. The first such number is 4 because it is composite and the sum of its divisors (1, 2 and 4) is 7 and relatively prime to 4. The next composite number is 6 whose sum of divisors (1, 2, 3 and 6) is 12. However, since 12 and 6 are not relatively prime it is not Duffinian. The next composite number is 8 with divisors 1, 2, 4 and 8 whose sum is 15 and thus relatively prime to 8. Thus the first two Duffinian numbers are 4 and 8.

The first Duffinian numbers up to 247 are:

4, 8, 9, 16, 21, 25, 27, 32, 35, 36, 39, 49, 50, 55, 57, 63, 64, 65, 75, 77, 81, 85, 93, 98, 100, 111, 115, 119, 121, 125, 128, 129, 133, 143, 144, 155, 161, 169, 171, 175, 183, 185, 187, 189, 201, 203, 205, 209, 215, 217, 219, 221, 225, 235, 237, 242, 243, 245, 247

Since there are 59 numbers listed above, their density is 59/247 or nearly 24%. Thus they are more common than the prime numbers that occur with a frequency of around 18% in the same interval. There is an infinity of such numbers. As explained in Numbers Aplenty:

Any number of the form  \( p^k \)  where  \( p \)  is prime and  \( k \ge 2 \)  is Duffinian, since  \( \sigma(p^k)=1+p+p^2+\cdots+p^k \)  cannot be divisible by \(p \).

Of course, every prime \( p \) is relatively prime to the sum of its divisors \( p+1 \) but by definition the prime numbers are excluded. However, OEIS A014567 lists numbers \(n \) such that \( n \) and \( \sigma(n) \) are relatively prime, without \( n \) needing to be composite. Such numbers are sometimes called sigma primes and of course include all the primes. OEIS A003624 lists the Duffinian numbers.

Numbers Aplenty goes on to say that:

P. Heichelheim proved that there exists a run of five consecutive Duffinian numbers starting at 202605639573839041, and that there cannot exist a longer such run. 

Rose Mary Zbiek has proved that every even Duffinian number is either a square or twice a square. 

The smallest 3 × 3 magic square whose entries are consecutive Duffinian numbers is:

It's a simple enough matter in SAGE to determine whether or not a number is Duffinian. Here is the code (with "gcd" standing for "greatest common divisor"):

number=25387
if gcd(number,sigma(number,1))==1:
    print number, "is Duffinian"
else:
    print number, "is not Duffinian"

The input 25387 produces the output: 25387 is Duffinian