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%I #66 Feb 19 2020 10:33:30
%S 6,120,126,2520,2640,30240,32640,37800,37926,55440,685440,758520,
%T 831600,2600640,5533920,6917400,9102240,10281600,11377800,16687440,
%U 152182800,206317440,250311600,475917120,866829600,1665709920,1881532800,2082137400,2147450880
%N Numbers that are 3-imperfect.
%C The new terms come from the paper by Zhou and Zhu. This sequence also contains n = 9223372034707292160 = 2^31*3*5*17*257*65537, which has the product of five Fermat primes (A019434). For this n, n/3 is a 2-imperfect number (A127725). - _T. D. Noe_, Apr 03 2009
%C From _M. F. Hasler_, Feb 13 2020: (Start)
%C By definition, n is k-imperfect iff n = k*A206369(n).
%C So a k-imperfect number is always a multiple of k, and up to the first odd 3-imperfect number (larger than 10^49, if it exists, see Zhou & Zhu (2009)), all terms are a multiple of 6. (End)
%H Michel Marcus, <a href="/A127726/b127726.txt">Table of n, a(n) for n = 1..75</a> (terms < 10^20) (terms 1 to 35 from Donovan Johnson)
%H Michel Marcus, <a href="/A127726/a127726.txt">More 3-imperfect numbers</a>, (includes 15 terms beyond a(75)), Nov 23 2017.
%H László Tóth, <a href="http://arxiv.org/abs/1111.4842">A survey of the alternating sum-of-divisors function</a>, arXiv:1111.4842 [math.NT], 2011-2014.
%H Weiyi Zhou and Long Zhu, <a href="http://www.emis.de/journals/INTEGERS/papers/j1/j1.Abstract.html">On k-imperfect numbers</a>, INTEGERS: Electronic Journal of Combinatorial Number Theory, 9 (2009), #A01.
%e 6 = 2*3, so A206369(6) = (2 - 1)(3 - 1) = 2 = 6 / 3, so 6 is a term.
%e 120 = 2^3 * 3 * 5, (8-4+2-1)*(3-1)*(5-1) = 40 = 120 / 3, so 120 is another term.
%t okQ[n_] := 3 Sum[d*(-1)^PrimeOmega[n/d], {d, Divisors[n]}] == n;
%t For[k = 3, k < 10^6, k = k + 3, If[okQ[k], Print[k]]] (* _Jean-François Alcover_, Feb 01 2019 *)
%o (PARI) isok(n) = 3*sumdiv(n, d, d*(-1)^bigomega(n/d)) == n; \\ _Michel Marcus_, Oct 28 2017
%o (PARI) select( {is_A127726(n)=A206369(n)*3==n}, [1..10^5]*6) \\ _M. F. Hasler_, Feb 13 2020
%Y Cf. A127724 (k-imperfect), A127725 (2-imperfect), A206369 (the rho function).
%K nonn
%O 1,1
%A _T. D. Noe_, Jan 25 2007
%E Extended by _T. D. Noe_, Apr 03 2009
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