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A002827 Unitary perfect numbers: usigma(n)-n = n.
(Formerly M4268 N1783)
6, 60, 90, 87360, 146361946186458562560000 (list; graph; refs; listen; history; text; internal format)



d is a unitary divisor of n if gcd(d,n/d)=1; usigma(n) is their sum (A034448).

The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - Paul Muljadi, Oct 10 2005

It is not known if a(6) exists. - N. J. A. Sloane, Jul 27 2015


R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.

F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=1..5.

A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.

M. V. Subbarao, Letter to N. J. A. Sloane, Feb 18 1974

M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta, 3 (No. 1, 1972), 22-26.

G. Villemin's Almanac of Numbers, Nombres Unitairement Parfaits

C. R. Wall, Letter to P. Hagis, Jr., Jan 13 1972

C. R. Wall, The fifth unitary perfect number, Canad. Math. Bull., 18 (1975), 115-122.

C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312-316.

Eric Weisstein's World of Mathematics, Unitary Perfect Number.

Wikipedia, Unitary perfect number


Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.

146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.


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 *)


(PARI) is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ Charles R Greathouse IV, Aug 01 2016


Cf. A034460, A034448, A057447.

Sequence in context: A007358 A322486 A007357 * A137498 A250070 A036283

Adjacent sequences:  A002824 A002825 A002826 * A002828 A002829 A002830




N. J. A. Sloane.



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Last modified January 19 15:53 EST 2019. Contains 319307 sequences. (Running on oeis4.)