Amicable Numbers Research Articles

A note on Chowla’s function

Iterates of a number-theoretic function, defined by L ( n ) = σ ( n ) − ( 1 + n ) L(n) = \sigma (n) - (1 + n) , are investigated empirically, for n ≦ 10 5 n \leqq {10^5} . This search has yielded 9 reduced amicable pairs.

Unitary amicable numbers

Two positive integers are said to be unitary amicable if the sum of the unitary divisors of each is equal to their sum. In this paper a table of such numbers is given, and some theorems concerning them are proved.

Lower Bounds for Relatively Prime Amicable Numbers of Opposite Parity

Lower Bounds for Relatively Prime Amicable Numbers of Opposite Parity

Amicable Numbers and Their Distribution

Amicable Numbers and Their Distribution

On Amicable and Sociable Numbers

An exhaustive search has yielded 236 amicable pairs of which the lesser number is smaller than ${10^8}$, 57 pairs being new. It has also yielded 9 new sociable groups of order 10 or less, of which the lesser number is smaller than ${6.10^7}$; the 9 sociable groups are all of order 4. The sequence of iterates of the function $s(n) = \sigma (n) - n$ starting with 276 has also been extended to 119 terms.

The Discovery of Amicable Numbers

The Discovery of Amicable Numbers

Amicable numbers and their distribution

All the amicable pairs whose smaller member is less than108{10^8}are now known. Using this data a conjecture is made about the distribution of amicable pairs. A result on "crowds" (sociable numbers of order 3) is also given.

On amicable and sociable numbers

An exhaustive search has yielded 236 amicable pairs of which the lesser number is smaller than108{10^8}, 57 pairs being new. It has also yielded 9 new sociable groups of order 10 or less, of which the lesser number is smaller than6.107{6.10^7}; the 9 sociable groups are all of order 4. The sequence of iterates of the functions(n)=σ(n)−ns(n) = \sigma (n) - nstarting with 276 has also been extended to 119 terms.

Lower bounds for relatively prime amicable numbers of opposite parity.

Whether or not a pair of relatively prime amicable numbers exists is an open question. In this paper it is proved that if m m and n n are a pair of relatively prime amicable numbers of opposite parity then m n mn is greater than 10 121 {10^{121}} and m m and n n are each greater than 10 60 {10^{60}} .

Relatively Prime Amicable Numbers of Opposite Parity

Relatively Prime Amicable Numbers of Opposite Parity

Relatively Prime Amicable Numbers of Opposite Parity

Relatively Prime Amicable Numbers of Opposite Parity

On Relatively Prime Odd Amicable Numbers

Whether or not a relatively prime pair of amicable numbers exists is still an open question. In this paper some necessary conditions for $m$ and $n$ to be a pair of odd relatively prime amicable numbers are proved. In particular, lower bounds for $m$, $n$, $mn$ and the number of prime divisors of $mn$ are established. The arguments are based on an extensive case study carried out on the CDC 6400 at the Temple University Computing Center.

On Divisibility by Nine of the Sums of Even Amicable Pairs

On Divisibility by Nine of the Sums of Even Amicable Pairs

On divisibility by nine of the sums of even amicable pairs

Most known even amicable pairs have sums divisible by nine [9]. The general form of the exceptions to the rule of divisibility by nine (Gardner’s rule) is deduced and the results expressed in the form of a theorem. A computer search based on a corollary to the theorem is described and six new exceptions to Gardner’s rule are found.

On relatively prime odd amicable numbers

Whether or not a relatively prime pair of amicable numbers exists is still an open question. In this paper some necessary conditions for m m and n n to be a pair of odd relatively prime amicable numbers are proved. In particular, lower bounds for m m , n n , m n mn and the number of prime divisors of m n mn are established. The arguments are based on an extensive case study carried out on the CDC 6400 at the Temple University Computing Center.

More Amicable Numbers

More Amicable Numbers

Correction: Amicable Numbers with Opposite Parity

Correction: Amicable Numbers with Opposite Parity

Amicable numbers and the bilinear diophantine equation

2. Amicable Numbers. Method I. An amicable pair, (n1, n2) is defined by (4) o(n1) = o-((n2) = nli + n2 where adenotes the divisor sum function [1]. If we let (5) ni=Apq and n2= Br where p, q are primes relatively prime to some number A, and r is a prime relatively prime to a number B, (4) gives (6) S(p+1)(q+1) = T(r+1) =Apq+Br whereS = o-(A), T=o-(B). Solving for r and eliminating r gives (7) r = S(p + 1)(q + 1)/T -1 and (8) [AT-S(T-B)]pq-S(T-B)(p + q) = BT + S(T-B). Substituting in (2) we obtain (9) ad + bc = T[ABT + (A -B)(T-B)S]. Our procedure now is to find all factor pairs, M, N, of T[ABT + (A B) (T B) S] such that MN = T[ABT + (A B) (T -B) S]. For any such pair we solve the linear equations, (lOa) [AT-S(T-B)]p= N + S(T-B), (lOb) [AT-S(T-B)]q= M + S(T-B) given by (3) observing that (lOb) has a solution if and only if (lOa) has a solution

More amicable numbers

A recent paper [1] listed all pairs of amicable numbers below 106. The following extension contains all amicable pairs with the lesser number between 106 and 107. Pairs marked with an asterisk are not found in the lists given by Escott [2], Poulet [3], and Garcia [4]. However, our referee has kindly pointed out that 22 of these pairs appear in a paper by Lee [5] which was in the process of publication. The 16 pairs marked t appear to be new, and complete the list of 108 pairs less than 107.

Amicable Numbers and the Bilinear Diophantine Equation

Amicable Numbers and the Bilinear Diophantine Equation