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%I #84 May 12 2023 07:01:12
%S 12,18,20,24,40,56,88,104,196,224,234,368,464,650,992,1504,1888,1952,
%T 3724,5624,9112,11096,13736,15376,15872,16256,17816,24448,28544,30592,
%U 32128,77744,98048,122624,128768,130304,174592,396896,507392,521728,522752,537248
%N 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 Named near-perfect numbers by sequence author.
%C Union of this sequence and A005820 is A153501.
%C 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 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 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 Conjecture 2. All entries are even. [Proved to be false, see below. (Ed.)]
%C 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 Conjecture 4. If a suitable divisor for an even entry is odd, then it is a Mersenne prime (A000043).
%C 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 The only odd term in this sequence < 2*10^12 is 173369889. - _Donovan Johnson_, Feb 15 2012
%C 173369889 remains only odd term up to 1.4*10^19. - _Peter J. C. Moses_, Mar 05 2012
%C 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 Donovan Johnson, <a href="/A181595/b181595.txt">Table of n, a(n) for n = 1..200</a>
%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 Yanbin Li and Qunying Liao, <a href="https://doi.org/10.4134/JKMS.2015.52.4.751">A class of new near-perfect numbers</a>, J. Korean Math. Soc. 52 (2015), No. 4, pp. 751-763.
%H Paul Pollack and Vladimir Shevelev, <a href="https://doi.org/10.1016/j.jnt.2012.06.008">On perfect and near-perfect numbers</a>, J. Number Theory 132 (2012), pp. 3037-3046. <a href="http://arxiv.org/abs/1011.6160">arXiv preprint</a>, arXiv:1011.6160 [math.NT], 2010-2012.
%H X.-Z. Ren, Y.-G. Chen, <a href="http://dx.doi.org/10.1017/S0004972713000178">On near-perfect numbers with two distinct prime factors</a>, Bulletin of the Australian Mathematical Society, No 3 (2013), available on CJO2013. doi:10.1017/S0004972713000178.
%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.
%e The abundance of 12 is A033880(12) = 4, which is a proper divisor of 12, so 12 is in the sequence.
%p q:= n-> (t-> t>0 and t<n and irem(n, t)=0)(numtheory[sigma](n)-2*n):
%p select(q, [$1..600000])[]; # _Alois P. Heinz_, May 11 2023
%t Select[Range[550000], 0 < (d = DivisorSigma[1, #] - 2*#) < # && Divisible[#, d] &] (* _Amiram Eldar_, May 12 2023 *)
%o (PARI) is_A181595(n)=my(d=sigma(n)-2*n); (d>0) && (d<n) && !(n%d);
%o for(n=1,1e6,is_A181595(n)&&print1(n",")) \\ _M. F. Hasler_, Apr 14 2012; corrected by _Michel Marcus_, May 12 2023
%Y Cf. A000396, A005101, A153501, A005820.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Nov 01 2010
%E Definition shortened, entries checked by _R. J. Mathar_, Nov 17 2010
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