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.

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

Offset: 1

Author

Vladimir Shevelev, Nov 01 2010

Keywords

• nonn

Comments

Named near-perfect numbers by sequence author.

Union of this sequence and A005820 is A153501.

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

173369889 remains only odd term up to 1.4*10^19. - Peter J. C. Moses, Mar 05 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

Examples

			The abundance of 12 is A033880(12) = 4, which is a proper divisor of 12, so 12 is in the sequence.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..200
• Hùng Việt Chu, Divisibility of Divisor Functions of Even Perfect Numbers, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
• Yanbin Li and Qunying Liao, A class of new near-perfect numbers, J. Korean Math. Soc. 52 (2015), No. 4, pp. 751-763.
• 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.
• 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.
• M. Tang, X. Z. Ren and M. Li, On near-perfect and deficient-perfect numbers, Colloq. Math. 133 (2013), 221-226.

Crossrefs

Cf. A000396, A005101, A153501, A005820.

Programs

Maple

q:= n-> (t-> t>0 and tAlois P. Heinz, May 11 2023

Mathematica

Select[Range[550000], 0 < (d = DivisorSigma[1, #] - 2*#) < # && Divisible[#, d] &] (* Amiram Eldar, May 12 2023 *)

PARI

is_A181595(n)=my(d=sigma(n)-2*n); (d>0) && (dA181595(n)&&print1(n","))  \\ M. F. Hasler, Apr 14 2012; corrected by Michel Marcus, May 12 2023

Extensions

Definition shortened, entries checked by R. J. Mathar, Nov 17 2010