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A181595
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Abundant numbers n for which the abundance d = sigma(n) - 2*n is a proper divisor, that is, 0 < d < n and d | n.
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26
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12, 18, 20, 24, 40, 56, 88, 104, 196, 224, 234, 368, 464, 650, 992, 1504, 1888, 1952, 3724, 5624, 9112, 11096, 13736, 15376, 15872, 16256, 17816, 24448, 28544, 30592, 32128, 77744, 98048, 122624, 128768, 130304, 174592, 396896, 507392, 521728, 522752, 537248
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OFFSET
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1,1
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COMMENTS
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Named near-perfect numbers by sequence author.
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.
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).
Conjecture 1: For every k>=1, there exist infinitely many entries m for which (2^k)|m and sigma(m)-2^k = 2*m.
Conjecture 2. All entries are even. [Proved to be false, see below. (Ed.)]
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.
Conjecture 4. If a suitable divisor for an even entry is odd, then it is a Mersenne prime (A000043).
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
The only odd term in this sequence < 2*10^12 is 173369889. - Donovan Johnson, Feb 15 2012
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
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LINKS
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EXAMPLE
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The abundance of 12 is A033880(12) = 4, which is a proper divisor of 12, so 12 is in the sequence.
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MAPLE
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q:= n-> (t-> t>0 and t<n and irem(n, t)=0)(numtheory[sigma](n)-2*n):
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MATHEMATICA
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Select[Range[550000], 0 < (d = DivisorSigma[1, #] - 2*#) < # && Divisible[#, d] &] (* Amiram Eldar, May 12 2023 *)
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PROG
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(PARI) is_A181595(n)=my(d=sigma(n)-2*n); (d>0) && (d<n) && !(n%d);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Definition shortened, entries checked by R. J. Mathar, Nov 17 2010
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STATUS
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approved
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