A181595 -id:A181595 - cp's OEIS frontend

A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 is prime.

12, 88, 1888, 32128, 521728, 8378368, 34359083008, 549753192448, 2251799645913088, 9223372026117357568, 2361183241263023915008, 2596148429267413634121263069790208, 2722258935367507707522529418717050175488

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

All these numbers are in A181595 because their abundance is 4, a proper divisor of m.

Cf. A059608, A156560, A181595, A181701, A000396, A181703, A045769

Rest[2^(#-1) (2^#-5)&/@(Round[N[Log[#+5]/Log[2]]]&/@Select[Table[2^t-5,{t,120}],PrimeQ])] (* Harvey P. Dale, Dec 16 2010 *)

571728 replaced with 521728 by R. J. Mathar, Dec 05 2010

A214408 Abundant numbers for which the abundance is not a divisor.

Original entry on oeis.org

30, 36, 42, 48, 54, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 114, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 198, 200, 204, 208, 210, 216, 220, 222, 228, 240, 246, 252, 258, 260, 264, 270, 272, 276, 280, 282

Offset: 1

Alonso del Arte, Jul 16 2012

nonn

The abundance of a number is sigma(n) - 2n (A033880).
Most of these numbers are pseudoperfect (A005835), but more than one proper divisor is left out of the sum.
The first odd term is 945, the second is 1575. The smallest odd abundant number not in this sequence is 173369889, found by Donovan Johnson. Peter J. C. Moses has verified all other odd abundant numbers up to 1.4 * 10^19 have an abundance that is not a proper divisor.
Almost all multiples of 6 are in this sequence. Given a prime p > 3, the abundance of 6p works out to 12, but 6p is not divisible by 4, though it is by 2 and by 3. The abundance of 12p is 16p + 28, and clearly that is not a divisor of 12p. Multiples of 6 with more prime factors will have abundances that are greater than the largest proper divisor by greater margins still.

			The abundance of 36 is 19, but 19 is not a divisor of 36, hence 36 is in the sequence.
The abundance of 40 is 10, which is a divisor of 40, hence 40 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A033880, A005835, A181595, A006037 (weird numbers, a subset).

filter:= proc(n) local b; b:= numtheory:-sigma(n) - 2*n; b > 0 and n mod b <> 0 end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 27 2016

Select[A005101, Not[MemberQ[Divisors[#], DivisorSigma[1, #] - 2#]] &]
anumQ[n_]:=Module[{a=DivisorSigma[1,n]-2n},a>0&&!Divisible[n,a]]; Select[Range[300],anumQ] (* Harvey P. Dale, Dec 23 2016 *)

is(n) = {my(ab = sigma(n) - 2*n); ab > 0 && n % ab;} \\ Amiram Eldar, Apr 07 2024

Corrected by Harvey P. Dale, Dec 23 2016

A181601 Numbers m with divisor 32 | m and abundance sigma(m)-2*m = 32.

Original entry on oeis.org

992, 28544, 122624, 507392, 537248, 698528, 791264, 1081568, 1279136, 2279072, 5029184, 307801856, 623799776, 712023296, 11196261056, 14809750016, 34355412992, 59640734144, 340536203264, 637707589184, 1091487733184, 1473169206272, 1709840369984, 2526522709184

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

A subsequence of A175989. - R. J. Mathar, Nov 04 2010

Amiram Eldar, Table of n, a(n) for n = 1..45 (calculated using the b-file at A175989)

Cf. A000203, A005101, A005820, A045768, A097498, A118372, A153501, A175989, A181595, A181596, A181597, A181598, A181599.

Select[32Range[1000000],DivisorSigma[1,#]-2#==32&] (* Harvey P. Dale, Aug 16 2011 *)

Definition rephrased, a(5)-a(11) appended - R. J. Mathar, Nov 04 2010

a(12)-a(24) from Donovan Johnson, Dec 08 2011

A181705 Numbers of the form 2^(t-1)*(2^t-9), where 2^t-9 is prime.

Original entry on oeis.org

56, 368, 128768, 2087936, 8589344768, 2199013818368, 36893488108764397568, 904625697166532776746648320380374279912262923807289020860114158381451706368

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

Subsequence of A181595.
(Proof: Let m=2^(t-1)*(2^t-9) be the entry. By the multiplicative property of the sigma-function, sigma(m)=(2^t-1)*(2^t-8).
The abundance sigma(m)-2*m is therefore 8, and since all t involved are >=4, 8 is a divisor of m because 8 divides 2^(t-1).)

Cf. A059610, A181595, A181701, A000396, A181703, A181704

2^(#-1) (2^#-9)&/@Select[Range[3,130],PrimeQ[2^#-9]&] (* Harvey P. Dale, Oct 24 2011 *)

Edited by R. J. Mathar, Sep 12 2011

A303359 Bi-unitary near-perfect numbers: bi-unitary abundant numbers k such that the abundance d = bsigma(k) - 2*k is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

24, 40, 56, 80, 88, 104, 120, 224, 360, 432, 672, 832, 992, 1008, 1296, 1456, 1504, 1584, 1888, 1952, 2016, 2160, 2800, 3800, 5624, 5800, 7424, 7616, 9112, 10080, 11096, 13736, 15872, 16256, 17816, 22848, 24448, 28544, 30592, 32128, 33728, 51136, 62464, 66368

Offset: 1

Amiram Eldar and Michael De Vlieger, Apr 22 2018

nonn

The bi-unitary version of A181595.

			24 is in the sequence since the sum of its bi-unitary divisors is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and 60 - 2*24 = 12 is a bi-unitary divisor of 24.

Amiram Eldar, Table of n, a(n) for n = 1..183

Cf. A153501, A181595, A188999, A292982, A303358.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; biunitaryDivisorQ[ div_, n_] := If[Mod[#2, #1]==0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]]&, {#1, #2/#1}]]==1, False]& @@{div, n}; aQ[n_] := Module[{d=bsigma[n]-2n},If[d<=0, False,biunitaryDivisorQ[d,n]]]; s={}; Do[If[ aQ[n], AppendTo[s,n] ], {n, 1, 10000}]; s

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
isok(n) = my(divs = biudivs(n), sig = vecsum(divs)); (sig > 2*n) && vecsearch(divs, sig - 2*n); \\ Michel Marcus, Apr 27 2018

A305616 Near 2-hyperperfect numbers: numbers k such that sigma(k) - 3*k/2 - 1/2 is a proper divisor of k.

Original entry on oeis.org

15, 63, 147, 171, 207, 627, 663, 1023, 1647, 1971, 2975, 6399, 18063, 19359, 27639, 40215, 48895, 58563, 78819, 95511, 114231, 133595, 134871, 145915, 147455, 163539, 168507, 172287, 188067, 529983, 680859, 795639, 1207359, 1238571, 1553499, 1588491, 2049219

Offset: 1

Amiram Eldar, Jun 06 2018

nonn

Supersequence of A063906.

A combination of the notions of 2-hyperperfect numbers (A007593) and near-perfect numbers (A181595).

			15 is in the sequence since sigma(15) = 24 and 24 - 3*15/2 - 1/2 = 1 is a proper divisor of 15.

Amiram Eldar, Table of n, a(n) for n = 1..70
Bhabesh Das and Helen K. Saikia, Identities for Near and Deficient Hyperperfect Numbers, Indian Journal in Number Theory, Vol. 3 (2016), pp. 124-134.

Cf. A000203, A007593, A063906, A181595, A305617.

aQ[n_] := Module[{d=DivisorSigma[1, n]-3n/2-1/2}, d>0 && d!=n && IntegerQ[d] && Divisible[n,d]]; Select[Range[1000000], aQ]

isok(n) = (n % 2) && (k = sigma(n) - (3*n+1)/2) && (k>0) && !(n % k) && (k != n); \\ Michel Marcus, Jun 07 2018

A341475 2-near-perfect numbers.

Original entry on oeis.org

12, 18, 24, 30, 36, 40, 48, 54, 56, 66, 80, 84, 90, 96, 112, 126, 132, 156, 176, 198, 200, 208, 220, 270, 280, 304, 352, 364, 380, 448, 550, 570, 594, 690, 736, 882, 910, 918, 928, 945, 992, 1026, 1040, 1120, 1216, 1372, 1376, 1488, 1638, 1696, 1722, 1782

Offset: 1

Jeffrey Shallit, Feb 13 2021

nonn

A number n is k-near-perfect if n is the sum of all but k of the proper divisors of n. Perfect numbers are 0-near-perfect and sequence A181595 lists the 1-near-perfect numbers.

			48 is 2-near-perfect because its proper divisors are {1, 2, 3, 4, 6, 8, 12, 16, 24} and 48 = 1+2+3+4+6+8+24.

Michael S. Branicky, Table of n, a(n) for n = 1..2000
Vedant Aryan, Dev Madhavani, Savan Parikh, Ingrid Slattery, and Joshua Zelinsky, On 2-Near Perfect Numbers, arXiv:2310.01305 [math.NT], 2023.
Hùng Việt Chu, Divisibility of Divisor Functions of Even Perfect Numbers, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046; also on arXiv, arXiv:1011.6160 [math.NT], 2010-2012.

Cf. A000396, A181595.

from sympy import divisors
def ok(n):
  proper_divs = divisors(n)[:-1]
  s = sum(proper_divs)
  if s - 3 < n: return False
  if s - sum(proper_divs[-2:]) > n: return False
  for i, c1 in enumerate(proper_divs[:-1]):
    if s - c1 - proper_divs[i+1] < n: return False
    if s - c1 - n in proper_divs[i+1:]: return True
  return False
def aupto(limit): return [m for m in range(1, limit+1) if ok(m)]
print(aupto(1782)) # Michael S. Branicky, Feb 21 2021

A181599 Numbers m with divisor 16 | m and abundance sigma(m)-2*m = 16.

Original entry on oeis.org

1504, 30592, 4526272, 8353792, 361702144, 1081850752, 1845991216, 2146926592, 21818579968, 34357510144, 228354264064, 549746900992, 2169800814592, 8796057370624, 24038405705152, 80952364306432, 140737345748992, 2737658648639872, 23810602502029312, 36979953305070592

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

Cf. A097498, A141547, A181595, A181596, A181597, A181598, A118372, A045768, A000396, A005101, A153501, A005820.

A008598 INTERSECT A141547. - R. J. Mathar, Nov 04 2010

Definition rephrased - R. J. Mathar, Nov 04 2010
a(9)-a(13) from Donovan Johnson, Dec 08 2011
a(14)-a(20) from the b-file at A141547 added by Amiram Eldar, Aug 03 2024

A181706 Numbers of the form 2^(t-1)*(2^t-17), where 2^t-17 is prime.

Original entry on oeis.org

1504, 30592, 8353792, 2146926592, 34357510144, 549746900992, 8796057370624, 140737345748992, 9223372000347553792, 2361183240850707054592, 9671406556879650002305024, 154742504910523000781012992

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

All entries are near-perfect numbers (A181595). The proof follows as in A181705, but this time the abundance is 16.

Cf. A059611, A181595, A181701, A000396, A181703, A181704, A181705.

A182142 Abundance d = sigma(N) - 2*N = A033880(N) of numbers N = A153501(n), i.e., N has d > 0 as divisor.

Original entry on oeis.org

4, 3, 2, 12, 10, 8, 4, 2, 120, 7, 56, 78, 8, 2, 2, 672, 32, 16, 4, 2, 532, 152, 136, 8, 68, 31, 992, 128, 8, 64, 32, 16, 4, 8, 128, 32, 8, 2, 43648, 2528, 32, 4, 2, 523776, 32, 2272, 32, 32, 127, 16256, 32, 32, 4, 536, 8, 32, 8, 52, 16, 32, 41044, 64

Offset: 1

M. F. Hasler, Apr 14 2012

nonn

It is conjectured that only powers of 2 can occur more than once.

Thanks to Amiram Eldar, reference to A181595 in the definition has been corrected to A153501 (which does include triperfect numbers, as required here, in contrast to A181595 where these are excluded). - M. F. Hasler, Sep 11 2019

M. F. Hasler, Table of n, a(n) for n = 1..200, Sep 11 2019

Cf. A033880, A153501, A181595.

f182142(n)={my(d=sigma(n)-2*n); d>0 && !(n%d) && return(d)} /* Note: This is A033880(n)*is_A153501(n), neither A182142 nor is_A182142. */
for(n=1,1e6,(t=f182142(n))&&print1(t","))

Equals A033880 o A153501.

cp's OEIS Frontend

A181704 Numbers m=2^(t-1)*(2^t-5), where 2^t-5 is prime.

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A214408 Abundant numbers for which the abundance is not a divisor.

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A181601 Numbers m with divisor 32 | m and abundance sigma(m)-2*m = 32.

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A181705 Numbers of the form 2^(t-1)*(2^t-9), where 2^t-9 is prime.

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A303359 Bi-unitary near-perfect numbers: bi-unitary abundant numbers k such that the abundance d = bsigma(k) - 2*k is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

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A305616 Near 2-hyperperfect numbers: numbers k such that sigma(k) - 3*k/2 - 1/2 is a proper divisor of k.

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A341475 2-near-perfect numbers.

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A181599 Numbers m with divisor 16 | m and abundance sigma(m)-2*m = 16.

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