A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k. (Formerly M4268 N1783)

6, 60, 90, 87360, 146361946186458562560000

Offset

1,1

Comments

d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) 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

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

Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - Ivan N. Ianakiev, Jan 20 2020

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.

H. A. M. Frei, Über unitar perfekte Zahlen, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.

Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.

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

Formula

If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

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 *)

Prog

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

Crossrefs

Cf. A034460, A034448, A057447.

Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.

Gives the positions of ones in A327159.

Sequence in context: A324707 A007357 A327158 * A331111 A324199 A137498

Adjacent sequences: A002824 A002825 A002826 * A002828 A002829 A002830

Keyword

nonn,nice,hard

Author

N. J. A. Sloane

Status

approved