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A083874 Numbers n such that sigma(n) + tau(n) = 2n. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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, 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, 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]

a(25) > 10^13. - Giovanni Resta, Jun 28 2013

a(36) > 2 * 10^18. - Hiroaki Yamanouchi, Aug 28 2018

1983709193244506192 is a term. - Hiroaki Yamanouchi, Aug 28 2018

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

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..35

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.

PROG

(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

CROSSREFS

Cf. A007503, A105330, A105331.

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

Sequence in context: A056076 A117133 A238226 * A105331 A017946 A083424

Adjacent sequences:  A083871 A083872 A083873 * A083875 A083876 A083877

KEYWORD

nonn

AUTHOR

Jason Earls, Jun 18 2003

EXTENSIONS

1082640256 found by Vladeta Jovovic, Sep 14 2004

More terms from David Wasserman, Dec 03 2004

More terms from Farideh Firoozbakht, Apr 27 2005

a(23) from Bill McEachen, 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, Dec 29 2008

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

STATUS

approved

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Last modified September 24 04:10 EDT 2020. Contains 337316 sequences. (Running on oeis4.)