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%I A181595 #84 May 12 2023 07:01:12
%S A181595 12,18,20,24,40,56,88,104,196,224,234,368,464,650,992,1504,1888,1952,
%T A181595 3724,5624,9112,11096,13736,15376,15872,16256,17816,24448,28544,30592,
%U A181595 32128,77744,98048,122624,128768,130304,174592,396896,507392,521728,522752,537248
%N A181595 Abundant numbers n for which the abundance d = sigma(n) - 2*n is a proper divisor, that is, 0 < d < n and d | n.
%C A181595 Named near-perfect numbers by sequence author.
%C A181595 Union of this sequence and A005820 is A153501.
%C A181595 Every even perfect number n = 2^(p-1)*(2^p-1), p and 2^p-1 prime, of A000396 generates three entries: 2*n, 2^p*n and (2^p-1)*n.
%C A181595 Every number M=2^(t-1)*P, where P is a prime of the form 2^t-2^k-1, is an entry for which (2^k)|M and sigma(M)-2^k=2*M (see A181701).
%C A181595 Conjecture 1: For every k>=1, there exist infinitely many entries m for which (2^k)|m and sigma(m)-2^k = 2*m.
%C A181595 Conjecture 2. All entries are even. [Proved to be false, see below. (Ed.)]
%C A181595 Conjecture 3. If the suitable (according to the definition) divisor d of an entry is not a power of 2, then it is not suitable divisor for any other entry.
%C A181595 Conjecture 4. If a suitable divisor for an even entry is odd, then it is a Mersenne prime (A000043).
%C A181595 If Conjectures 3 and 4 are true, then an entry with odd suitable divisor has the form 2^(p-1)*(2^p-1)^2, where p and 2^p-1 are primes. - _Vladimir Shevelev_, Nov 08 2010 to Dec 16 2010
%C A181595 The only odd term in this sequence < 2*10^12 is 173369889. - _Donovan Johnson_, Feb 15 2012
%C A181595 173369889 remains only odd term up to 1.4*10^19. - _Peter J. C. Moses_, Mar 05 2012
%C A181595 These numbers are obviously pseudoperfect (A005835) since they are equal to the sum of all the proper divisors except the one that is the same as the abundance. - _Alonso del Arte_, Jul 16 2012
%H A181595 Donovan Johnson, Table of n, a(n) for n = 1..200
%H A181595 Hùng Việt Chu, Divisibility of Divisor Functions of Even Perfect Numbers, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
%H A181595 Yanbin Li and Qunying Liao, A class of new near-perfect numbers, J. Korean Math. Soc. 52 (2015), No. 4, pp. 751-763.
%H A181595 Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046. arXiv preprint, arXiv:1011.6160 [math.NT], 2010-2012.
%H A181595 X.-Z. Ren, Y.-G. Chen, On near-perfect numbers with two distinct prime factors, Bulletin of the Australian Mathematical Society, No 3 (2013), available on CJO2013. doi:10.1017/S0004972713000178.
%H A181595 M. Tang, X. Z. Ren and M. Li, On near-perfect and deficient-perfect numbers, Colloq. Math. 133 (2013), 221-226.
%e A181595 The abundance of 12 is A033880(12) = 4, which is a proper divisor of 12, so 12 is in the sequence.
%p A181595 q:= n-> (t-> t>0 and t0) && (d