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A002827 |
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Unitary perfect numbers: numbers k such that usigma(k) - k = k. (Formerly M4268 N1783) |
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45 |
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OFFSET |
1,1
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COMMENTS |
d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
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
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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).
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LINKS |
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FORMULA |
If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020
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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.
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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 *)
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PROG |
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CROSSREFS |
Gives the positions of ones in A327159.
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KEYWORD |
nonn,nice,hard
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AUTHOR |
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STATUS |
approved
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