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%I #19 Jan 12 2019 02:27:36
%S 1258418761414,1276686130498,1286096593354,1290188098942,
%T 1306261870882,1321049741038,1338795185146,1350625481098,
%U 1359498202882,1365723585502,1367261834038,1371277504834,1372962401386,1373062247098,1373771709754,1374112095298,1374709701094
%N Numbers whose sum of proper divisors is equal to 690100611194.
%C The number 690100611194 is the 49th term of A283157. That is, no even number below it has more preimages under the sum-of-proper-divisors function. Up to 2^40, this is the even number with the greatest number of preimages. As of May 22 2017, this is the largest known even number with the greatest number of preimages.
%C There are exactly 139 terms 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="/A287262/b287262.txt">Table of n, a(n) for n = 1..139</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) = 1258418761414, because it is the smallest number whose sum of proper divisors is equal to 690100611194: 1 + 2 + 31 + 62 + 20297076797 + 40594153594 + 629209380707 = 690100611194.
%Y Cf. A001065, A283156, A283157, A287233, A287238, A287247, A287251.
%K fini,full,nonn
%O 1,1
%A _Anton Mosunov_, May 22 2017
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