%I #31 Sep 28 2018 23:20:34

%S 48,56,192,248,252,328,448,496,768,1016,1792,2032,3240,6462,7936,8128,

%T 11616,11808,17412,20538,49152,65528,114688,131056,507904,524224,

%U 786432,1048568,1835008,2080768,2096896,2097136,3145728,4194296,7340032,8126464,8388544,8388592,32505856,33292288,33554176,33554368,133169152,134217472

%N Non-unitary amicable numbers.

%C 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.

%C 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.

%C No other pair below 10^9.

%C 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

%H Charles R Greathouse IV, <a href="/A259037/b259037.txt">Table of n, a(n) for n = 1..48</a>

%H Steve Ligh and Charles R. Wall, <a href="https://www.fq.math.ca/Scanned/25-4/ligh.pdf">Functions of Nonunitary Divisors</a>, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary divisor">Unitary divisor</a>

%e 48 and 56 are in the sequence, as sigma(48)-usigma(48) = 56 and sigma(56)-usigma(56) = 48.

%o (PARI) A048146(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1)

%o is(n)=my(k=A048146(n)); k>1 && k!=n && A048146(k)==n \\ _Charles R Greathouse IV_, Jun 17 2015

%Y Subsequence of A013929.

%Y Cf. A000043, A034448, A048146, A259038, A259039.

%K nonn

%O 1,1

%A _Mauro Fiorentini_, Jun 17 2015