A088912 a(n) = smallest m such that sigma(m) = (n+1/2)*m.

2, 24, 4320, 8910720, 17116004505600, 170974031122008628879954060917200710847692800, 12749472205565550032020636281352368036406720997031277595140988449695952806020854579200000

Offset: 1

Author

Farideh Firoozbakht, Nov 29 2003

Keywords

• hard
• nonn

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

Examples

			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.
		

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

Crossrefs

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

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).

Programs

Mathematica

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

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