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%I #57 Aug 26 2023 20:31:57
%S 1,2,4,8,10,16,32,44,64,128,136,152,184,256,512,752,884,1024,2048,
%T 2144,2272,2528,4096,8192,8384,12224,16384,17176,18632,18904,32768,
%U 32896,33664,34688,49024,63248,65536,85936,106928,116624,117808,131072,262144,524288,526688,527872,531968,556544,589312,599072,654848,709784
%N Deficient-perfect numbers: Deficient numbers n such that n/(2n-sigma(n)) is an integer.
%C Every power of 2 is part of this sequence, with 2n - sigma(n) = 1.
%C Any odd element of this sequence must be a perfect square with at least four distinct prime factors (Tang and Feng). The smallest odd element of this sequence is a(74) = 9018009 = 3^2 * 7^2 * 11^2 * 13^2, with 2n - sigma(n) = 819.
%C a(17) = 884 = 2^2 * 13 * 17 is the smallest element with three distinct prime factors.
%C For all n > 1 in this sequence, 5/3 <= sigma(n)/n < 2. - _Charles R Greathouse IV_, Apr 15 2016
%H Giovanni Resta, <a href="/A271816/b271816.txt">Table of n, a(n) for n = 1..273</a> (terms < 2*10^12)
%H BYU Computational Number Theory Group, <a href="https://arxiv.org/abs/2006.10697">Odd, spoof perfect factorizations</a>, arXiv:2006.10697 [math.NT], 2020.
%H Jose A. B. Dris, <a href="https://doi.org/10.7546/nntdm.2018.24.3.62-67">Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017), pp. 12-20, <a href="https://arxiv.org/abs/1610.01868">arXiv preprint</a>, arXiv:1610.01868 [math.NT], 2016.
%H Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, <a href="https://doi.org/10.7546/nntdm.2019.25.1.199-205">A note on the OEIS sequence A228059</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
%H Hùng Việt Chu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Chu/chu26.html">Divisibility of Divisor Functions of Even Perfect Numbers</a>, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
%H Cui-Fang Sun and Zhao-Cheng He, <a href="https://arxiv.org/abs/1908.04932">On odd deficient-perfect numbers with four distinct prime divisors</a>, arXiv:1908.04932 [math.NT], 2019.
%H Judy Holdener and Emily Rachfal, <a href="https://doi.org/10.1080/00029890.2019.1584515">Perfect and Deficient Perfect Numbers</a>, The American Mathematical Monthly, Vol. 126, No. 6 (2019), pp. 541-546.
%H M. Tang, X. Z. Ren, and M. Li, <a href="http://dx.doi.org/10.4064/cm133-2-8">On Near-Perfect and Deficient-Perfect Numbers</a>, Colloq. Math. 133 (2013), 221-226.
%H M. Tang and M. Feng, <a href="http://dx.doi.org/10.1017/S0004972714000082">On Deficient-Perfect Numbers</a>, Bull. Aust. Math. Soc. 90 (2014), 186-194.
%F 2^k is always an element of this sequence.
%F If 2^(k+1) + 2^t - 1 is an odd prime and t <= k, then n = 2^k(2^(k+1) + 2^t - 1) is deficient-perfect with 2n - sigma(n) = 2^t. In fact, these are the only terms with two distinct prime factors. (Tang et al.)
%e When n = 1, 2, 4, 8, 2n - sigma(n) = 1.
%e When n = 10, sigma(10) = 18 and so 2*10 - 18 = 2, which divides 10.
%p q:= k-> (s-> s>0 and irem(k, s)=0)(2*k-numtheory[sigma](k)):
%p select(q, [$1..500000])[]; # _Alois P. Heinz_, Aug 26 2023
%t ok[n_] := Block[{d = DivisorSigma[1, n]}, d < 2*n && Divisible[n, 2*n - d]]; Select[Range[10^5], ok] (* _Giovanni Resta_, Apr 14 2016 *)
%o (PARI) isok(n) = ((ab = (sigma(n)-2*n))<0) && (n % ab == 0); \\ _Michel Marcus_, Apr 15 2016
%Y Cf. A000203, A005100, A033879.
%Y Deficient analog of A153501. Contains A000079.
%K nonn
%O 1,2
%A _Carlo Francisco E. Adajar_, Apr 14 2016
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