A000396 -id:A000396 - cp's OEIS frontend

A007357 Infinitary perfect numbers.

6, 60, 90, 36720, 12646368, 22276800, 126463680, 4201148160, 28770487200, 287704872000, 1446875426304, 2548696550400, 14468754263040, 590325173932032, 3291641594841600, 8854877608980480, 32916415948416000

Offset: 1

N. J. A. Sloane

nonn

Numbers N whose sum of infinitary divisors equals 2*N: A049417(N)=2*N. - Joerg Arndt, Mar 20 2011

6 is the only infinitary perfect number which is also perfect number (A000396). 6 is also the only squarefree infinitary perfect number. - Vladimir Shevelev, Mar 02 2011

			Let n=90. Its unique expansion over distinct terms of A050376 is 90=2*5*9. Thus the infinitary divisors of 90 are 1,2,5,9,10,18,45,90. The number of such divisors is 2^3, i.e., the cardinality of the set of all subsets of the set {2,5,9}. The sum of such divisors is (2+1)*(5+1)*(9+1)=180 and the sum of proper such divisors is 90=n. Thus 90 is in the sequence. - _Vladimir Shevelev_, Mar 02 2011

G. L. Cohen, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
David Moews, A database of aliquot cycles - Known infinitary perfect numbers (together with unitary perfect and e-perfect ones).
Jan Munch Pedersen, Known infinitary perfect numbers. [BROKEN LINK]
Eric Weisstein's World of Mathematics, Infinitary Perfect Number.

Cf. A129656 (infinitary abundant), A129657 (infinitary deficient).

isA007357 := proc(n)
    A049417(n) = 2*n ;
    simplify(%) ;
end proc:
for n from 1 do
    if isA007357(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Oct 05 2017

infiPerfQ[n_] := 2n == Total[If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m&])]]];
For[n = 6, True, n += 6, If[infiPerfQ[n], Print[n]]] (* Jean-François Alcover, Feb 08 2021 *)

{n: A049417(n) = 2*n}. - R. J. Mathar, Mar 18 2011

a(n)==0 (mod 6). Thus there are no odd infinitary perfect numbers. - Vladimir Shevelev, Mar 02 2011

More terms from Eric W. Weisstein, Jan 27 2004

A083206 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 17, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 14, 0, 0, 0, 1, 0, 13, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 2, 0, 1

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

a(n)=0 for deficient numbers n (A005100), but the converse is not true, as 18 is abundant (A005101) and a(18)=0, see A083211;
a(n)=1 for perfect numbers n (A000396), see A083209 for all numbers with a(n)=1;
records: A083213(k)=a(A083212(k)).
In order that a(n)>0, the sum of divisors of n must be even by definition: a(n) = half the number of partitions of A000203(n)/2 into divisors of n, see formula. [Reinhard Zumkeller, Jul 10 2010]

			a(24)=3: 1+2+3+4+8+12=6+24, 1+3+6+8+12=2+4+24, 4+6+8+12=1+2+3+24.

Antti Karttunen, Table of n, a(n) for n = 1..101632 (terms 1..800 from Reinhard Zumkeller, terms 801..10000 from T. D. Noe).
Reinhard Zumkeller, Illustration of initial terms

Cf. A083208 [= a(A083207(n))], A083211, A000005, A000203, A082729, A378446 (inverse Möbius transform), A378449.
Cf. A083207 (positions of terms > 0), A083210 (positions of 0's), A083209 (positions of 1's), A378652 (of 2's).
Cf. also A033630, A065205, A058377, A325806.

a[n_] := (s = DivisorSigma[1, n]; If[Mod[s, 2] == 1, 0, f[n, s/2, 2]]); f[n_, m_, k_] := f[n, m, k] = If[k <= m, f[n, m, k+1] + f[n, m-k, k+1]*Boole[Mod[n, k] == 0], Boole[m == 0]]; Array[a, 105] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)

A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); }; \\ Antti Karttunen, Dec 02 2024, after Ilya Gutkovskiy

a(n) = if sigma(n) mod 2 = 1 then 0 else f(n,sigma(n)/2,2), where sigma=A000203 and f(n,m,k) = if k<=m then f(n,m,k+1)+f(n,m-k,k+1)*0^(n mod k) else 0^m, cf. A033630, also using f. [Reinhard Zumkeller, Jul 10 2010]

a(n) is half the coefficient of x^0 in Product_{d|n} (x^d + 1/x^d). - Ilya Gutkovskiy, Feb 04 2024

A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

Original entry on oeis.org

8, 32, 512, 8192, 33554432, 8589934592, 137438953472, 2305843009213693952, 2658455991569831745807614120560689152, 191561942608236107294793378393788647952342390272950272

Offset: 1

Omar E. Pol, Apr 13 2008

nonn

Sum of n-th even perfect number and n-th even superperfect number.

Also, sum of n-th perfect number and n-th superperfect number, if there are no odd perfect and odd superperfect numbers, then the n-th perfect number is the difference between a(n) and the n-th superperfect number (see A135652, A135653, A135654 and A135655).

			a(5) = 33554432 because A000043(5) = 13 and 2^(2*13 - 1) = 2^25 = 33554432.
Also, if there are no odd perfect and odd superperfect numbers then we can write a(5) = A000396(5) + A019279(5) = A000396(5) + A061652(5) = 33554432.

Amiram Eldar, Table of n, a(n) for n = 1..15
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000079, A000396, A019279, A061645, A061652, A133033, A135652, A135653, A135654, A135655, A139286, A139294, A139307.

Cf. A000043, A000668, A072868.

2^(2 * MersennePrimeExponent[Range[10]] - 1) (* Amiram Eldar, Oct 17 2024 *)

a(n) = 2^(2*A000043(n) - 1). Also, a(n) = 2^A133033(n), if there are no odd perfect numbers. Also, a(n) = A000396(n) + A019279(n), if there are no odd perfect and odd superperfect numbers. Also, a(n) = A000396(n) + A061652(n), if there are no odd perfect numbers, then we can write: perfect number A000396(n) = a(n) - A061652(n).

a(n) = A061652(n)*(A000668(n)+1) = A061652(n)*A072868(n). - Omar E. Pol, Apr 13 2008

A174090 Powers of 2 and odd primes; alternatively, numbers that cannot be written as a sum of at least three consecutive positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 17, 19, 23, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 256

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 07 2010, and Omar E. Pol, Feb 24 2014

From Omar E. Pol, Feb 24 2014: (Start)
Also the odd noncomposite numbers (A006005) and the powers of 2 with positive exponent, in increasing order.
If a(n) is composite and a(n) - a(n-1) = 1 then a(n-1) is a Mersenne prime (A000668), hence a(n-1)*a(n)/2 is a perfect number (A000396) and a(n-1)*a(n) equals the sum of divisors of a(n-1)*a(n)/2.
If a(n) is even and a(n+1) - a(n) = 1 then a(n+1) is a Fermat prime (A019434). (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
Nieuw Archief voor Wiskunde, Problems/UWC, Problem C, Vol. 5/6, No. 2.

Numbers not in A111774.
Equals A000079 UNION A065091.
Equals A067133 \ {6}.
Cf. A000040, A000203, A000396, A000668, A006005, A019434, A092506.
Cf. also A138591, A174069, A174070, A174071.

N:= 300: # to get all terms <= N
S:= {seq(2^i,i=0..ilog2(N))} union select(isprime,{ 2*i+1 $ i=1..floor((N-1)/2) }):
sort(convert(S,list)); # Robert Israel, Jun 18 2015

a[n_] := Product[GCD[2 i - 1, n], {i, 1, (n - 1)/2}] - 1;
Select[Range[242], a[#] == 0 &] (* Gerry Martens, Jun 15 2015 *)

list(lim)=Set(concat(concat(1,primes(lim)), vector(logint(lim\2,2),i,2^(i+1)))) \\ Charles R Greathouse IV, Sep 19 2024

select( {is_A174090(n)=isprime(n)||n==1<M. F. Hasler, Oct 24 2024

from sympy import primepi
def A174090(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+(0 if x<=1 else 1-primepi(x))-x.bit_length())
    return bisection(f,n,n) # Chai Wah Wu, Sep 19 2024

a(n) ~ n log n. - Charles R Greathouse IV, Sep 19 2024

This entry is the result of merging an old incorrect entry and a more recent correct version. N. J. A. Sloane, Dec 07 2015

A090748 Numbers k such that 2^(k+1) - 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126, 520, 606, 1278, 2202, 2280, 3216, 4252, 4422, 9688, 9940, 11212, 19936, 21700, 23208, 44496, 86242, 110502, 132048, 216090, 756838, 859432, 1257786, 1398268, 2976220, 3021376, 6972592, 13466916, 20996010, 24036582, 25964950, 30402456, 32582656

Offset: 1

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 03 2004

nonn

Perfect numbers A000396(n) = 2^A133033(n) - 2^a(n), assuming there are no odd perfect numbers. - Omar E. Pol, Feb 24 2008
Number of proper divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 28 2008
Base 2 logarithm of n-th even superperfect number A061652(n). Also base 2 logarithm of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Apr 11 2008
Number of 0's in binary expansion of n-th even perfect number (see A135650). - Omar E. Pol, May 04 2008

			1 is in the sequence because 2^2 - 1 = 3 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

a(n) = A000043(n) - 1. A000043 is the main entry for this sequence.

Cf. A000396, A133033, A000668, A019279, A061652, A135650, A051027.

[n: n in [1..5*10^3] |IsPrime(2^(n+1)-1)]; // Vincenzo Librandi, Jul 28 2016

Select[Range[0, 10^4], PrimeQ[2^(# + 1) - 1] &] (* Vincenzo Librandi, Jul 28 2016 *)

is(n)=ispseudoprime(2^(n+1)-1) \\ Charles R Greathouse IV, Aug 21 2016

a(n) = A000043(n) - 1.

2^(a(n) + 1) = A051027(2^a(n)). - Juri-Stepan Gerasimov, Aug 21 2016 [corrected by Jerzy R Borysowicz, Feb 26 2025]

Edited, corrected and extended by Robert G. Wilson v, Feb 09 2004
a(39) from Omar E. Pol, Jan 23 2009
a(40)-a(44) from Ivan Panchenko, Apr 11 2018

A139256 Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n).

Original entry on oeis.org

12, 56, 992, 16256, 67100672, 17179738112, 274877382656, 4611686016279904256, 5316911983139663489309385231907684352, 383123885216472214589586756168607276261994643096338432

Offset: 1

Omar E. Pol, Apr 22 2008

nonn

Also, twice perfect numbers, if there are no odd perfect numbers.
If there are no odd perfect numbers, essentially the same as A065125. - R. J. Mathar, May 23 2008
The sum of reciprocals of even divisors of a(n) equals 1. Proof: Let n = (2^m - 1)*2^m where 2^m - 1 is a Mersenne prime. The sum of reciprocals of even divisors of n is s1 + s2 where: s1 = 1/2 + 1/4 + ... + 1/2^m = (2^m - 1)/2^m and s2 = s1/(2^m - 1) => s1 + s2 = 1. - Michel Lagneau, Jul 17 2013

			a(3) = 992 because the third Mersenne prime A000668(3) is 31 and 31*(31+1) = 31*32 = 992.
a(3) = 992 because the sum of the divisors of the third perfect number is 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 992. - _Omar E. Pol_, Dec 05 2016
From _Omar E. Pol_, Aug 13 2021: (Start)
Illustration of initial terms in which a(n) is represented as the sum of the divisors of the n-th even perfect number P(n).
-------------------------------------------------------------------------
  n  P(n) a(n)  Diagram:   1                                           2
-------------------------------------------------------------------------
                           _                                           _
                          | |                                         | |
                          | |                                         | |
                       _ _| |                                         | |
                      |    _|                                         | |
                 _ _ _|  _|                                           | |
  1    6   12   |_ _ _ _|                                             | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                             _ _ _ _ _| |
                                                            |  _ _ _ _ _|
                                                            | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|
                                             _ _ _| |
                                            |  _ _ _|
                                            | |
                                            | |
                                            | |
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  2   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) equals the area (also the number of cells) in the n-th diagram.
For n = 3, P(3) = 496 and a(3) = 992, the diagram is too large to include here. To draw that diagram note that the lengths of the line segments of the smallest Dyck path are [248, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 248] and the lengths of the line segments of the largest Dyck path are [249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 249]. For a definition of these numbers related to partitions into consecutive parts see A237591. (End)

Walter A. Kehowski, Power-spectral Numbers, ResearchGate (2024); also available at vixra.org.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000203, A000396, A000668, A001065, A065125, A139257, A237591, A237593, A245092.

DeleteCases[2 Map[(# (# + 1))/2 &, Select[2^Range[100] - 1, PrimeQ]], k_ /; OddQ@ k] (* Michael De Vlieger, Dec 05 2016, after Harvey P. Dale at A000396 *)

a(n) = A000668(n)*(A000668(n)+1).
a(n) = 2*A000396(n), if there are no odd perfect numbers.
a(n) = A000203(A000396(n)) = A001065(A000396(n)) + A000396(n), assuming there are no odd perfect numbers. - Omar E. Pol, Dec 04 2016

More terms from Omar E. Pol, Jun 07 2012

A324201 a(n) = A062457(A000043(n)) = prime(A000043(n))^A000043(n), where A000043 gives the exponent of the n-th Mersenne prime.

Original entry on oeis.org

9, 125, 161051, 410338673, 925103102315013629321, 1271991467017507741703714391419, 49593099428404263766544428188098203, 165163983801975082169196428118414326197216835208154294976154161023

Offset: 1

Antti Karttunen, Feb 18 2019

nonn

If there are no odd perfect numbers, then the terms give all solutions n > 1 to A323244(n) = 0.

Conversely, if these are all numbers k > 1 that satisfy A323244(k) = 0 (which can be proved if one can show, for example, that no number in A007916 can satisfy the equation), then no odd perfect numbers exist. See also A336700. - Antti Karttunen, Jan 12 2024

Amiram Eldar, Table of n, a(n) for n = 1..12

Cf. A000043, A000396, A005940, A007916, A062457, A323244, A324200.
Subsequence of A001597.
Cf. also A336700, A368989.

Prime[#]^#&/@MersennePrimeExponent[Range[8]] (* Harvey P. Dale, Mar 15 2024 *)

a(n) = A062457(A000043(n)).
A323244(a(n)) = 0.
a(n) = A005940(1+A000396(n)). [Provided no odd perfect numbers exist]

A175522 A000120-perfect numbers.

Original entry on oeis.org

2, 25, 95, 111, 119, 123, 125, 169, 187, 219, 221, 247, 289, 335, 365, 411, 415, 445, 485, 493, 505, 629, 655, 685, 695, 697, 731, 767, 815, 841, 871, 943, 949, 965, 985, 1003, 1139, 1207, 1241, 1261, 1263, 1273, 1343, 1387, 1465, 1469, 1507, 1513, 1529, 1563

Offset: 1

Vladimir Shevelev, Dec 03 2010

nonn

Let A(n), n>=1, be an infinite positive sequence.
We call a number n:
   A-deficient if Sum{d|n, d   A-abundant if Sum{d|n, d A(n),
and
   A-perfect if Sum{d|n, ddepending on the sum over the proper divisors of n.
The definition generalizes the standard nomenclature of deficient (A005100), abundant (A005101) and perfect numbers (A000396), which is recovered by setting A(n) = n = A000027(n).
Conjecture: if there exist infinitely many A-deficient numbers and infinitely many A-abundant numbers, then there exist infinitely many A-perfect numbers.
Note that the sequence contains squares of all Fermat primes larger than 3 (see A019434). [This would also hold for squares of any hypothetical Fermat primes after the fifth one, 65537. Comment clarified by Antti Karttunen, May 14 2015]
A192895(a(n)) = 0. - Reinhard Zumkeller, Jul 12 2011

			Proper divisors of 119 are 1,7,17. Since A000120(1)+A000120(7)+A000120(17)=A000120(119), then 119 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A175524 (deficient version), A175526 (abundant version), A000120, A000396.
Subsequence of A257691 (non-abundant version).
Positions of zeros in A192895.
Cf. A019434, A253204.

import Data.List (elemIndices)
a175522 n = a175522_list !! (n-1)
a175522_list = map (+ 1) $ elemIndices 0 a192895_list
-- Reinhard Zumkeller, Jul 12 2011

binw[n_] := DigitCount[n, 2, 1]; Select[Range[1500], binw[#] == DivisorSum[#, binw[#1] &]/2 &] (* Amiram Eldar, Dec 14 2020 *)

is(n)=sumdiv(n,d,hammingweight(d))==2*hammingweight(n) \\ Charles R Greathouse IV, Jan 28 2016

from sympy import divisors
def A000120(n): return bin(n).count('1')
def aupto(limit):
  alst = []
  for m in range(1, limit+1):
    if A000120(m) == sum(A000120(d) for d in divisors(m)[:-1]): alst += [m]
  return alst
print(aupto(1563)) # Michael S. Branicky, Feb 25 2021

A000120 = lambda x: x.digits(base=2).count(1)
is_A175522 = lambda x: sum(A000120(d) for d in divisors(x)) == 2*A000120(x)
A175522 = filter(is_A175522, IntegerRange(1, 10**4))
# D. S. McNeil, Dec 04 2010

More terms from Amiram Eldar, Feb 18 2019

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.

Original entry on oeis.org

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

Vladimir Shevelev, Nov 01 2010

nonn

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

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

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.

Cf. A000396, A005101, A153501, A005820.

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

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

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

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

A064591 Nonunitary perfect numbers: k is the sum of its nonunitary divisors; i.e., k = sigma(k) - usigma(k).

Original entry on oeis.org

24, 112, 1984, 32512, 134201344, 34359476224, 549754765312

Offset: 1

Dean Hickerson, Sep 25 2001

There are no other terms up to 1.2*10^14.
If P (A000396) is an even perfect number, then 4*P is in the sequence. Are there any others?
If there are no terms of another form, the sequence goes on with 9223372032559808512 = 2^32 * A000668(8), 10633823966279326978618770463815368704 = 2^62 * A000668(9), 766247770432944429179173512337214552523989286192676864 = 2^90 * A000668(10), ... - Michel Lagneau, Jan 21 2015
Conjecture: let s0 be the sum of the inverses of the even divisors of a number n and s1 the sum of the inverses of the odd divisors of n; then n is in the sequence iff 2*s0-s1 = 1. - Michel Lagneau, Jan 21 2015
Ligh & Wall proved that 2^(p+1)*(2^p-1) is a term if p and 2^p-1 are primes, and that all the nonunitary perfect numbers below 10^6 are of this form. - Amiram Eldar, Sep 27 2018
If k is in the sequence and k = 2^m*p^a then k is of the form 4*P for an even perfect P. See the link to MathOverflow. - Joshua Zelinsky, Mar 07 2024

			The sum of the nonunitary divisors of 24 is 2 + 4 + 6 + 12 = 24.

Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.
MathOverflow, Are all nonunitary perfect numbers in the form 4k where k is an even perfect number?

Cf. A000203 (sigma), A034448 (usigma), A048146, A063870.

Cf. A064592, A064593, A064594, A064595, A064596, A064597, A064598, A064599.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]==n, Print[ n ] ] ]
Do[s0=0;s1=0;Do[d=Divisors[n][[i]];If[Mod[d,2]==0,s0=s0+1/d,s1=s1+1/d],{i,1,Length[Divisors[n]]}];If[2*s0-s1==1,Print[n]],{n,2,10^9,2}] (* Michel Lagneau, Jan 21 2015 *)

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A007357 Infinitary perfect numbers.

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A083206 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum.

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A139306 Ultraperfect numbers: a(n) = 2^(2*p - 1), where p is A000043(n).

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A174090 Powers of 2 and odd primes; alternatively, numbers that cannot be written as a sum of at least three consecutive positive integers.

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A090748 Numbers k such that 2^(k+1) - 1 is prime.

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A139256 Twice even perfect numbers. Also a(n) = M(n)*(M(n)+1), where M(n) is the n-th Mersenne prime A000668(n).

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A324201 a(n) = A062457(A000043(n)) = prime(A000043(n))^A000043(n), where A000043 gives the exponent of the n-th Mersenne prime.