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A271816 Deficient-perfect numbers: Deficient numbers n such that n/(2n-sigma(n)) is an integer. 1

%I

%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 M. Tang, X. Z. Ren, 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.

%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 Deficient analog of A153501. Contains A000079.

%K nonn

%O 1,2

%A _Carlo Francisco E. Adajar_, Apr 14 2016

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Last modified June 25 06:13 EDT 2017. Contains 288709 sequences.