1,2

Dihedral perfect numbers: degree n such that dihedral group D_{2n} has order equal to its total number of subgroups, i.e. A007503(n)=2n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 14 2004

There are no terms between the two numbers 283417216 & 537346048 that were found by Vladeta Jovovic. If 2^(m+1)+2m+1 is prime then 2^m*(2^(m+1)+2m+1) is in the sequence (see A105330 & A105331). This fact is a result of the following interesting theorem that I found (take k=0). Theorem: If m & k are integers and 2^(m+1)+2m+1-k is prime then 2^m*(2^(m+1)+2m+1-k) is a solution of the equation sigma(x)+ tau(x)=2x+k; the proof is easy. - Farideh Firoozbakht (mymontain(AT)yahoo.com), Apr 27 2005

The sequence is complete through a(23). Allowing n = 2^k pq, for primes p < q, it is fairly easy to discover that p must be in the range 2^(k+1) to 2^(k+2) and there is only one possible q for each p. Exhaustive search can be used to find the primes p. The sequence up to a(23) has all possible solutions for k=1, 3 and 7. There is one solution for k=10: 91707741184; one solution for k=12: 14451706793984; and three solutions for k=15: 2258918614925312, 1153167823398797312, 4611826823562493952. There are no other n = 2^k pq for k up to 26. - T. D. Noe (noe(AT)sspectra.com), Jun 19 2008

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 130, p. 44, Ellipses, Paris 2008.

J.-M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 723 pp. 93;308 Ellipses Paris 2004.

(PARI) (running start from entry 20+) forstep(k= 1082640256, 2000000000000, 2, if((numdiv(k)+sigma(k))==2*k, write("A83874.txt", k))) - Bill Mceachen (bmceache(AT)centralsan.dst.ca.us), Jun 14 2006

Cf. A007503, A105330, A105331.

Sequence in context: A084150 A056076 A117133 this_sequence A105331 A017946 A083424

Adjacent sequences: A083871 A083872 A083873 this_sequence A083875 A083876 A083877

nonn,more

Jason Earls (zevi_35711(AT)yahoo.com), Jun 18 2003

1082640256 found by Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 14 2004

More terms from David Wasserman (wasserma(AT)spawar.navy.mil), Dec 03 2004

More terms from Farideh Firoozbakht (mymontain(AT)yahoo.com), Apr 27 2005

a(23) from Bill Mceachen (bmceache(AT)centralsan.dst.ca.us), Jun 14 2006, who reports that he has searched the full range through a(23).

91707741184 confirmed to be the 24th term by Donovan Johnson (donovan.johnson(AT)yahoo.com), Dec 29 2008