

A088912


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


4




OFFSET

1,1


COMMENTS

2 is the only number m such that sigma(m)=1.5*m.
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
If the Riemann hypothesis (RH) is true then Robin's theorem (Guy Robin, 1984) implies that the nth term of this sequence is greater than exp(exp((n+1/2)/exp(gamma))) where gamma=0.5772156649... is the EulerMascheroni 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
From Gerard P. Michon, Jul 04 2009: (Start)
An upper bound for a(7) is provided by a 97digit 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 286digit integer of abundancy 17/2 (3.30181...10^285) equal to x/17, where x is the smallest known number of abundancy 9 (a 287digit 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 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
These are the least hemiperfects of abundancy n + 1/2.  Walter Nissen, Aug 17 2010
On Jul 24 2010, Michel Marcus found a 191digit 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


REFERENCES

Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187213. [From Gerard P. Michon, Jun 10 2009]


LINKS

Table of n, a(n) for n=1..6.
G. P. Michon, Multiplicative functions: Abundancy = sigma(n)/n [From Gerard P. Michon, Jun 10 2009]
G. P. Michon, Multiperfect and hemiperfect integers [From Gerard P. Michon, Jun 10 2009]
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 11/2 [From Gerard P. Michon, Aug 06 2009]
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 13/2 [From Gerard P. Michon, Aug 06 2009]
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 15/2 [From Gerard P. Michon, Aug 06 2009]
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 17/2 [From Walter Nissen, Aug 17 2010]
Walter Nissen, Abundancy : Some Resources
Wikipedia, Hemiperfect number
Wikipedia, Riemann hypothesis [From Washington Bomfim, Oct 30 2008]


EXAMPLE

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.


MATHEMATICA

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


CROSSREFS

Cf. A007539, A000396, A005820, A027687, A046060, A046061.
Cf. A159907 (hemiperfect numbers: halfintegral abundancy), A141643 (abundancy = 5/2), A055153 (abundancy = 7/2), A141645 (abundancy = 9/2), A159271 (abundancy = 11/2), A160678 (abundancy = 13/2).
Sequence in context: A000794 A159907 A242484 * A342573 A203465 A055462
Adjacent sequences: A088909 A088910 A088911 * A088913 A088914 A088915


KEYWORD

hard,more,nonn


AUTHOR

Farideh Firoozbakht, Nov 29 2003


EXTENSIONS

a(5)a(6) from Robert Gerbicz, Apr 19 2009
Crossreferences from Gerard P. Michon, Jun 10 2009
Edited by M. F. Hasler, Mar 17 2013


STATUS

approved



