OFFSET
1,1
COMMENTS
A pair of integers x and y is called non-unitary amicable if the sum of the non-unitary divisors of either one is equal to the other. Union of A259038 and A259039.
The sequence lists the non-unitary amicable numbers in increasing order. Note that the pairs x, y are not always adjacent to each other in the list. See also A259038 for the x's, A259039 for the y's. The first time a pair is not adjacent is x = 11616, y = 17412 which correspond to a(17) and a(19), respectively.
No other pair below 10^9.
Ligh & Wall showed that if p and q are different Mersenne exponents (A000043) (i.e., 2^p - 1 and 2^q - 1 are Mersenne primes), then 2^(p+1) * (2^q-1) and 2^(q+1) * (2^p-1) is a nonunitary amicable pair. They also found the pairs (252, 328), (3240, 6462), (11616, 17412), (11808, 20538), which are all the known pairs that are not based on Mersenne primes. - Amiram Eldar, Sep 27 2018
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..48
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor
EXAMPLE
48 and 56 are in the sequence, as sigma(48)-usigma(48) = 56 and sigma(56)-usigma(56) = 48.
PROG
(PARI) A048146(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mauro Fiorentini, Jun 17 2015
STATUS
approved