| 2 is the only number m such that sigma(m)=1.5*m.
The next term in this sequence is greater than 5*10^9.
A direct consequence of Robin's theorem is that a(6)>5E16, a(7)>1.898E29, a(8)>2.144E51, a(9)>9.877E89 and a(10)>6.023E157. [From W. Bomfim (webonfim(AT)bol.com.br), Oct 30 2008]
If the Riemann hypothesis (RH) is true then Robin's theorem (Guy Robin, 1984) implies that the n-th term of this sequence is greater than exp(exp((n+1/2)/exp(gamma))) where gamma=0.5772156649... is the Euler-Mascheroni constant (A001620). For the 6-th term (which is actually 1.7*10^44) this lower bound is 5.0*10^16. Similarly, if RH is true, the next term (7-th term) is at least 1.9*10^29 (and is probably more than 10^90 or so). [From Gerard P. Michon (g.michon(AT)att.net), Jun 10 2009]
Contribution from Gerard P. Michon (g.michon(AT)att.net), Jul 04 2009: (Start)
An upper bound for a(7) is provided by a 97-digit integer of abundancy 15/2 (5.71379...10^96) discovered by Michel Marcus on July 4, 2009. The factorization of that number is: 2^53 3^15 5^6 7^6 11^3 13 17 19^3 23 29 31 37 41 43 61 73 79 97 181 193 199 257 263 4733 11939 19531 21803 87211 262657
Similarly, an upper bound for a(8) is provided by a 286-digit integer of abundancy 17/2 (3.30181...10^285) equal to x/17, where x is the smallest known number of abundancy 9 (a 287-digit integer discovered by Fred W. Helenius in 1995). This is so because 17 happen to occur with multiplicity 1 in the factorization of x. (End)
A new upper bound for a(7) was found on August 15, 2009 by Michel Marcus who broke his own record by finding two "small" multiples of 2^35*3^20*5^5*7^6*11^2*13^2*17 that are of abundancy 15/2. The lower one (1.27494722...10^88) has only 89 digits. [From Gerard P. Michon (g.michon(AT)att.net), Aug 15 2009]
These are the least hemiperfects of abundancy n + 1/2. [From Walter Nissen (nissen(AT)gtcinternet.com), Aug 17 2010]
On July 24, 2010, Michel Marcus found a 191-digit integer of abundancy 17/2 (2.7172904...10^190) whose factorization starts with 2^81 3^29 5^9 7^10 11^4 13^3 17^2 19 23^2... This is the best upper bound to a(8) known so far. [From Gerard P. Michon (g.michon(AT)att.net), Aug 22 2010]
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