

A002827


Unitary perfect numbers: usigma(n)n = n.
(Formerly M4268 N1783)


11




OFFSET

1,1


COMMENTS

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


REFERENCES

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).


LINKS

Table of n, a(n) for n=1..5.
M. B. 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), 2226.
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), 115122.
C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312316.
Eric Weisstein's World of Mathematics, Unitary Perfect Number.
Wikipedia, Unitary perfect number


EXAMPLE

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.


MATHEMATICA

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 *)


CROSSREFS

Cf. A034460, A034448.
Cf. A002827, A057447.
Sequence in context: A189000 A007358 A007357 * A137498 A250070 A036283
Adjacent sequences: A002824 A002825 A002826 * A002828 A002829 A002830


KEYWORD

nonn,nice,hard


AUTHOR

N. J. A. Sloane.


STATUS

approved



