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%I #39 Aug 28 2018 10:46:32
%S 1,3,14,52,130,184,656,8648,12008,34688,2118656,33721216,40575616,
%T 59376256,89397016,99523456,134438912,150441856,173706136,283417216,
%U 537346048,1082640256,6801628304,91707741184,14451706793984,102898828936832,141573123151232
%N Numbers n such that sigma(n) + tau(n) = 2n.
%C 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
%C 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
%C 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
%C 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]
%C a(25) > 10^13. - _Giovanni Resta_, Jun 28 2013
%C a(36) > 2 * 10^18. - _Hiroaki Yamanouchi_, Aug 28 2018
%C 1983709193244506192 is a term. - _Hiroaki Yamanouchi_, Aug 28 2018
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 130, p. 44, Ellipses, Paris 2008.
%D J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 723 pp. 93;308 Ellipses Paris 2004.
%D J.-M. De Koninck & A. Mercier, 1001 Problems in Classical Number Theory, AMS, 2007.
%H Hiroaki Yamanouchi, <a href="/A083874/b083874.txt">Table of n, a(n) for n = 1..35</a>
%H David W. Jensen and Eric R. Bussian, <a href="http://www.jstor.org/stable/2686678">A Number-Theoretic Approach to Counting Subgroups of Dihedral Groups</a>, Two-Year College Math. Jnl., 23 (1992), 150-152.
%o (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
%Y Cf. A007503, A105330, A105331.
%Y Cf. A173168 (primes of the form 2^k+2k-1).
%K nonn
%O 1,2
%A _Jason Earls_, Jun 18 2003
%E 1082640256 found by _Vladeta Jovovic_, Sep 14 2004
%E More terms from _David Wasserman_, Dec 03 2004
%E More terms from _Farideh Firoozbakht_, Apr 27 2005
%E a(23) from _Bill McEachen_, Jun 14 2006, who reports that he has searched the full range through a(23)
%E 91707741184 confirmed to be the 24th term by _Donovan Johnson_, Dec 29 2008
%E a(25)-a(27) from _Hiroaki Yamanouchi_, Aug 28 2018
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