A259037 Non-unitary amicable numbers.

48, 56, 192, 248, 252, 328, 448, 496, 768, 1016, 1792, 2032, 3240, 6462, 7936, 8128, 11616, 11808, 17412, 20538, 49152, 65528, 114688, 131056, 507904, 524224, 786432, 1048568, 1835008, 2080768, 2096896, 2097136, 3145728, 4194296, 7340032, 8126464, 8388544, 8388592, 32505856, 33292288, 33554176, 33554368, 133169152, 134217472

Offset: 1

Author

Mauro Fiorentini, Jun 17 2015

Keywords

• nonn

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

Examples

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

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

Crossrefs

Subsequence of A013929.

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

Programs

PARI

A048146(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1)
is(n)=my(k=A048146(n)); k>1 && k!=n && A048146(k)==n \\ Charles R Greathouse IV, Jun 17 2015