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%I M4268 N1783 #63 Dec 26 2021 21:04:56
%S 6,60,90,87360,146361946186458562560000
%N Unitary perfect numbers: numbers k such that usigma(k) - k = k.
%C d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
%C The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - _Paul Muljadi_, Oct 10 2005
%C It is not known if a(6) exists. - _N. J. A. Sloane_, Jul 27 2015
%C Frei proved that if there is a unitary perfect number that is not divisible by 3, then it is divisible by 2^m with m >= 144, it has at least 144 distinct odd prime factors, and it is larger than 10^440. - _Amiram Eldar_, Mar 05 2019
%C Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - _Ivan N. Ianakiev_, Jan 20 2020
%D R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
%D F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H H. A. M. Frei, <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001:1978:33#105">Über unitar perfekte Zahlen</a>, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.
%H Takeshi Goto, <a href="http://doi.org/10.1216/rmjm/1194275935">Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers</a>, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
%H A. V. Lelechenko, <a href="http://taac.org.ua/files/a2014/proceedings/UA-2-Andrew%20Lelechenko-440.pdf">The Quest for the Generalized Perfect Numbers</a>, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
%H M. V. Subbarao, <a href="/A002827/a002827.pdf">Letter to N. J. A. Sloane, Feb 18 1974</a>
%H M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, <a href="http://www.math.ualberta.ca/~subbarao/documents/Subbarao_Cook_Newberry_Weber1972.pdf">On unitary perfect numbers</a>, Delta, 3 (No. 1, 1972), 22-26.
%H G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/ParfUnit.htm">Nombres Unitairement Parfaits</a>
%H C. R. Wall, <a href="/A002827/a002827_1.pdf">Letter to P. Hagis, Jr., Jan 13 1972</a>
%H C. R. Wall, <a href="http://dx.doi.org/10.4153/CMB-1975-021-9">The fifth unitary perfect number</a>, Canad. Math. Bull., 18 (1975), 115-122.
%H C. R. Wall, <a href="http://www.fq.math.ca/Scanned/25-4/wall1.pdf">On the largest odd component of a unitary perfect number</a>, Fib. Quart., 25 (1987), 312-316.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryPerfectNumber.html">Unitary Perfect Number.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_perfect_number">Unitary perfect number</a>
%F If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - _Amiram Eldar_, Jun 06 2020
%e Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.
%e 146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.
%t usnQ[n_]:=Total[Select[Divisors[n],GCD[#,n/#]==1&]]==2n; Select[Range[ 90000],usnQ] (* This will generate the first four terms of the sequence; it would take a very long time to attempt to generate the fifth term. *) (* _Harvey P. Dale_, Nov 14 2012 *)
%o (PARI) is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ _Charles R Greathouse IV_, Aug 01 2016
%Y Cf. A034460, A034448, A057447.
%Y Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.
%Y Gives the positions of ones in A327159.
%K nonn,nice,hard
%O 1,1
%A _N. J. A. Sloane_
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