A083874 Numbers k such that sigma(k) + tau(k) = 2k.

1, 3, 14, 52, 130, 184, 656, 8648, 12008, 34688, 2118656, 33721216, 40575616, 59376256, 89397016, 99523456, 134438912, 150441856, 173706136, 283417216, 537346048, 1082640256, 6801628304, 91707741184, 14451706793984, 102898828936832, 141573123151232

Offset: 1

Author

Jason Earls, Jun 18 2003

Keywords

• nonn

Comments

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, Oct 14 2004

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, Apr 27 2005

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, Jun 19 2008

Assuming n has the form 2^k pqr, for primes p < q < r, there are only a finite number of triples (p,q,r) possible for each k. For k=3, the sequence already has 89397016 and 173706136. For k=4, 6801628304 is the only solution. For k=7, the search finds 102898828936832, 141573123151232, 220346295412352, 619057492909952, and 3585817801980032. For k=9, 989473186649763328 is the only solution. There are no other solutions for k <= 13. - T. D. Noe, Feb 12 2010

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 130, p. 44, Ellipses, Paris 2008.
• J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 723 pp. 93;308 Ellipses Paris 2004.
• J.-M. De Koninck & A. Mercier, 1001 Problems in Classical Number Theory, AMS, 2007.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..40 (first 35 terms from Hiroaki Yamanouchi)
• David W. Jensen and Eric R. Bussian, A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups, Two-Year College Math. Jnl., 23 (1992), 150-152.

Crossrefs

Cf. A007503, A105330, A105331.

Cf. A173168 (primes of the form 2^k+2k-1).

Programs

PARI

/* running start from entry 20+*/ forstep(k= 1082640256, 2000000000000,2,if((numdiv(k)+sigma(k))==2*k,write("A083874.txt",k))) \\ Bill McEachen, Jun 14 2006

Extensions

a(11)-a(20) from David Wasserman, Dec 03 2004

a(21) from Farideh Firoozbakht, Apr 27 2005

a(22) from Vladeta Jovovic confirmed and a(23) added by Bill McEachen, Jun 14 2006

a(24) from T. D. Noe confirmed by Donovan Johnson, Dec 29 2008

a(25)-a(27) from Hiroaki Yamanouchi, Aug 28 2018

Edited by Max Alekseyev, Jul 30 2025