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Numbers whose sum of proper divisors is equal to 289697407994.
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%I #9 May 22 2017 12:11:35

%S 300629385430,331082751976,348602203870,539890623754,552235683634,

%T 556381352806,556523967562,557844696646,562012970938,569170200598,

%U 569518766962,573004430386,574282506778,575462269366,576199620754,577107726658,577305647026,577419601138

%N Numbers whose sum of proper divisors is equal to 289697407994.

%C The number 289697407994 is the 47th element of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function.

%C There are exactly 123 elements in the sequence.

%C In 2016, C. Pomerance proved that, for every e>0, the number of preimages is O_e(n^{2/3+e}).

%C Conjecture: there exists a positive real number k such that the number of preimages of an even number n is O((log n)^k).

%H Anton Mosunov, <a href="/A287247/b287247.txt">Table of n, a(n) for n = 1..123</a>

%H C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/aliquot.pdf">The first function and its iterates</a>, A Celebration of the Work of R. L. Graham, S. Butler, J. Cooper, and G. Hurlbert, eds., Cambridge U. Press, to appear.

%e a(1) = 300629385430, because it is the smallest number whose sum of proper divisors is equal to 289697407994: 1 + 2 + 5 + 10 + 11 + 22 + 55 + 110 + 2732994413 + 5465988826 + 13664972065 + 27329944130 + 30062938543 + 60125877086 + 150314692715 = 289697407994.

%Y Cf. A001065, A283156, A283157, A287233, A287238, A287251, A287262.

%K fini,full,nonn

%O 1,1

%A _Anton Mosunov_, May 22 2017