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 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
From Gerard P. Michon, 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 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 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
REFERENCES
Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
LINKS
G. P. Michon, Multiplicative functions: Abundancy = sigma(n)/n
G. P. Michon, Multiperfect and hemiperfect integers
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 11/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 13/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 15/2
G. P. Michon and M. Marcus, Hemiperfect numbers of abundancy 17/2
Walter Nissen, Abundancy : Some Resources
Wikipedia, Hemiperfect number
Wikipedia, Riemann hypothesis
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
KEYWORD
hard,nonn
AUTHOR
Farideh Firoozbakht, Nov 29 2003
EXTENSIONS
a(5)-a(6) from Robert Gerbicz, Apr 19 2009
Cross-references from Gerard P. Michon, Jun 10 2009
Edited by M. F. Hasler, Mar 17 2013
a(7) from Michel Marcus confirmed and added by Max Alekseyev, Jun 05 2025
STATUS
approved