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

6, 60, 90, 87360, 146361946186458562560000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

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

Examples

			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.
		

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).
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 147-148.

Links

• 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

Crossrefs

Cf. A034460, A034448, A057447.

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

Gives the positions of ones in A327159.

Programs

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 *)

PARI

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

Formula

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