

A259037


Nonunitary amicable numbers.


3



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
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OFFSET

1,1


COMMENTS

A pair of integers x and y is called nonunitary amicable if the sum of the nonunitary divisors of either one is equal to the other. Union of A259038 and A259039.
The sequence lists the nonunitary 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^q1) and 2^(q+1) * (2^p1) 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. 333338.
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)
is(n)=my(k=A048146(n)); k>1 && k!=n && A048146(k)==n \\ Charles R Greathouse IV, Jun 17 2015


CROSSREFS

Subsequence of A013929.
Cf. A000043, A034448, A048146, A259038, A259039.
Sequence in context: A080854 A255267 A345503 * A231469 A261546 A335938
Adjacent sequences: A259034 A259035 A259036 * A259038 A259039 A259040


KEYWORD

nonn


AUTHOR

Mauro Fiorentini, Jun 17 2015


STATUS

approved



