Search: id:A088912
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%I A088912
%S A088912 2,24,4320,8910720,17116004505600,
%T A088912 170974031122008628879954060917200710847692800
%N A088912 a(n) = smallest m such that sigma(m)=(n+1/2)*m.
%C A088912 2 is the only number m such that sigma(m)=1.5*m.
%C A088912 The next term in this sequence is greater than 5*10^9.
%C A088912 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]
%C A088912 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]
%C A088912 Contribution from Gerard P. Michon (g.michon(AT)att.net), Jul 04 2009: 
               (Start)
%C A088912 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 A088912 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 A088912 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]
%D A088912 Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese 
               de Riemann, J. Math. Pures Appl. 63 (1984), 187-213. [From Gerard 
               P. Michon (g.michon(AT)att.net), Jun 10 2009]
%H A088912 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">
               Abundancy : Some Resources </a>
%H A088912 Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_hypothesis#Growth_rates_of_multiplicative_functions"\
               >Riemann hypothesis</a> [From W. Bomfim (webonfim(AT)bol.com.br), 
               Oct 30 2008]
%H A088912 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">
               Multiplicative functions</a>: Abundancy = sigma(n)/n [From Gerard 
               P. Michon (g.michon(AT)att.net), Jun 10 2009]
%H A088912 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiperfect">
               Multiperfect and hemiperfect integers</a> [From Gerard P. Michon 
               (g.michon(AT)att.net), Jun 10 2009]
%H A088912 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 
               (g.michon(AT)att.net), Aug 06 2009]
%H A088912 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 
               (g.michon(AT)att.net), Aug 06 2009]
%H A088912 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 
               (g.michon(AT)att.net), Aug 06 2009]
%e A088912 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 A088912 a[n_] := (For[m=1, DivisorSigma[1, m]!=(n+1/2)m, m++ ];m); Do[Print[a[n]], 
               {n, 4}]
%Y A088912 Cf. A007539, A055153, A000396, A005820, A027687.
%Y A088912 A141643 (abundancy = 5/2), A141645 (abundancy = 9/2), A159271 (abundancy 
               = 11/2), A160678 (abundancy = 13/2), A159907 (half-integral abundancy, 
               "hemiperfect numbers"). [From Gerard P. Michon (g.michon(AT)att.net), 
               Jun 10 2009]
%Y A088912 Sequence in context: A059332 A000794 A159907 this_sequence A055462 A088600 
               A066120
%Y A088912 Adjacent sequences: A088909 A088910 A088911 this_sequence A088913 A088914 
               A088915
%K A088912 hard,more,nonn
%O A088912 1,1
%A A088912 Farideh Firoozbakht (f.firoozbakht(AT)sci.ui.ac.ir), Nov 29 2003
%E A088912 a(5)-a(6) from Robert Gerbicz (robert.gerbicz(AT)gmail.com), Apr 19 2009

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