%I #37 Nov 22 2021 12:43:58

%S 2,24,4320,8910720,17116004505600,

%T 170974031122008628879954060917200710847692800

%N a(n) = smallest m such that sigma(m) = (n+1/2)*m.

%C 2 is the only number m such that sigma(m)=1.5*m.

%C 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. - _Washington Bomfim_, Oct 30 2008

%C 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 6th term (which is actually 1.7*10^44) this lower bound is 5.0*10^16. Similarly, if RH is true, the next term (7th term) is at least 1.9*10^29 (and is probably more than 10^90 or so). - _Gerard P. Michon_, Jun 10 2009

%C From _Gerard P. Michon_, Jul 04 2009: (Start)

%C 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.

%C 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)

%C A new upper bound for a(7) was found on Aug 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. - _Gerard P. Michon_, Aug 15 2009

%C These are the least hemiperfects of abundancy n + 1/2. - _Walter Nissen_, Aug 17 2010

%C On Jul 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. - _Gerard P. Michon_, Aug 22 2010

%D Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213. [From _Gerard P. Michon_, Jun 10 2009]

%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">Multiplicative functions</a>: Abundancy = sigma(n)/n [From _Gerard P. Michon_, Jun 10 2009]

%H G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiperfect">Multiperfect and hemiperfect integers</a> [From _Gerard P. Michon_, Jun 10 2009]

%H G. P. Michon and M. Marcus, <a href="http://www.numericana.com/data/hpn11.htm">Hemiperfect numbers of abundancy 11/2</a> [From _Gerard P. Michon_, Aug 06 2009]

%H G. P. Michon and M. Marcus, <a href="http://www.numericana.com/data/hpn13.htm">Hemiperfect numbers of abundancy 13/2</a> [From _Gerard P. Michon_, Aug 06 2009]

%H G. P. Michon and M. Marcus, <a href="http://www.numericana.com/data/hpn15.htm">Hemiperfect numbers of abundancy 15/2</a> [From _Gerard P. Michon_, Aug 06 2009]

%H G. P. Michon and M. Marcus, <a href="http://www.numericana.com/data/hpn17.htm"> Hemiperfect numbers of abundancy 17/2</a> [From _Walter Nissen_, Aug 17 2010]

%H Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources </a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hemiperfect_number">Hemiperfect number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_hypothesis#Growth_rates_of_multiplicative_functions">Riemann hypothesis</a> [From _Washington Bomfim_, Oct 30 2008]

%e a(2)=24 because 1+2+3+4+6+8+12+24=2.5*24 and 24 is the earliest m such that sigma(m)=2.5*m.

%t a[n_] := (For[m=1, DivisorSigma[1, m]!=(n+1/2)m, m++ ];m); Do[Print[a[n]], {n, 4}]

%Y Cf. A007539, A000396, A005820, A027687, A046060, A046061.

%Y Cf. A159907 (hemiperfect numbers: half-integral abundancy), A141643 (abundancy = 5/2), A055153 (abundancy = 7/2), A141645 (abundancy = 9/2), A159271 (abundancy = 11/2), A160678 (abundancy = 13/2).

%K hard,more,nonn

%O 1,1

%A _Farideh Firoozbakht_, Nov 29 2003

%E a(5)-a(6) from _Robert Gerbicz_, Apr 19 2009

%E Cross-references from _Gerard P. Michon_, Jun 10 2009

%E Edited by _M. F. Hasler_, Mar 17 2013