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A083874
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Numbers n such that sigma(n) + tau(n) = 2n.
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11
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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
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OFFSET
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1,2
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COMMENTS
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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]
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REFERENCES
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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.
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LINKS
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PROG
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(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
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CROSSREFS
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Cf. A173168 (primes of the form 2^k+2k-1).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(23) from Bill McEachen, Jun 14 2006, who reports that he has searched the full range through a(23)
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STATUS
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approved
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