A181596 -id:A181596 - cp's OEIS frontend

A181598 Numbers m with divisor 8 | m and abundance sigma(m)-2*m = 8.

56, 368, 11096, 17816, 77744, 128768, 2087936, 2291936, 13174976, 35021696, 45335936, 381236216, 4856970752, 6800228816, 8589344768, 1461083549696, 1471763808896, 2199013818368, 19502341651712, 118123076415296, 933386556194816, 144141575952121856, 417857739454939136

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

a(19) > 10^13. - Giovanni Resta, Apr 02 2014

Cf. A008590, A088833, A097498, A181595, A181596, A181597, A118372, A045768, A000396, A005101, A153501, A005820.

isok(n) = !(n % 8) && (sigma(n) - 2*n == 8); \\ Michel Marcus, Feb 08 2016

A088833 INTERSECT A008590. - R. J. Mathar, Nov 04 2010

Definition rephrased by R. J. Mathar, Nov 04 2010
a(16)-a(17) from Donovan Johnson, Dec 08 2011
a(18) from Giovanni Resta, Apr 02 2014
a(19)-a(23) from the b-file at A088833 added by Amiram Eldar, Mar 11 2024

A181701 Near-perfect numbers (A181595) of the form 2^(t-1)*(2^t-2^k-1), where 2^t-2^k-1 is prime, k>=1, t>k.

Original entry on oeis.org

12, 20, 56, 88, 104, 368, 464, 992, 1504, 1888, 1952, 16256, 24448, 28544, 30592, 32128, 98048, 122624, 128768, 130304, 507392, 521728, 522752, 2087936, 7337984, 8124416, 8353792, 8378368, 8382464, 25161728, 67100672, 125820928, 132112384, 133685248, 134193152

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

There exist near-perfect numbers of the form 2^r*p, where p is prime, which are not in the sequence (e.g., 24,40,224). For given k, the smallest value of t gives sequence A181692.

Donovan Johnson, Table of n, a(n) for n = 1..1000
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:1011.6160
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. A181595, A181596, A000396, A181692.

s = Sort@ Flatten@ Table[p = (2^t - 2^k - 1); If[PrimeQ@ p, 2^(t - 1) p, Nothing], {t, 2, 14}, {k, t - 1}]; Select[Select[s, DivisorSigma[1, #] > 2 # &], MemberQ[Divisors@ #, DivisorSigma[1, #] - 2 #] &] (* Michael De Vlieger, Sep 23 2015, after Alonso del Arte at A181595 *)

mx=2^269*(2^270-2^122-1); v=vector(1000); n=0; for(k=1, 269, for(t=k+1, 270, p=2^t-2^k-1; m=2^(t-1)*p; if(m>mx, next(2)); if(isprime(p), n++; v[n]=m))); v=vecsort(v); for(n=1, 1000, write("b181701.txt", n " " v[n])) /* Donovan Johnson, May 24 2013 */

Edited, corrected, and extended by D. S. McNeil, Nov 18 2010

A181597 (N\{4})-perfect numbers, i.e., numbers m for which sigma(m)-4 = 2m, if 4|m, otherwise sigma(m) = 2m.

Original entry on oeis.org

6, 12, 88, 1888, 32128, 521728, 1848964, 8378368, 34359083008, 549753192448

Offset: 1

Vladimir Shevelev, Nov 01 2010, Nov 03 2010

Or union of {6}, near-perfect numbers m (cf. A181595) for which d(m)=4, and all odd perfect numbers (if they exist). Note that (N\{2})-perfect numbers are numbers for which sigma(m)-2=2m, if m is even, and sigma(m)=2m, if m is odd. They are all even numbers of A045768 and all odd perfect numbers (if they exist).

			88 is in the sequence since sigma(88) = 180 and 180 - 4 = 2*88.

Cf. A000396, A005101, A005820, A088832, A118372, A153501, A181595, A181596.

Invalid term removed and a(8)-a(10) from Donovan Johnson, Sep 14 2013

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

A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).

Original entry on oeis.org

24, 224, 15872, 1040384, 274844352512, 1125891316908032, 72057456598974464, 4951760154835678090382802944, 6129982163463555430774932117031404988667342368173719552

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..14

Subsequence of A181595 and A181702.

Cf. A000043, A000396, A147538, A181596, A000396, A181701, A181702.

With[{p = MersennePrimeExponent[Range[10]]}, 2^(2*p - 1)*(2^p - 1)] (* Amiram Eldar, Apr 29 2024 *)

a(n) = A147538(A000043(n)). - Amiram Eldar, Apr 29 2024

a(6)-a(9) from Amiram Eldar, Apr 29 2024

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

A181702 Numbers of the form 2^r*p with p an odd prime, which are in A181595 but not in A181701.

Original entry on oeis.org

24, 40, 224, 15872, 1040384

Offset: 1

Vladimir Shevelev, Nov 06 2010

a(6) > 1.85*10^11, if it exists. - Amiram Eldar, Apr 29 2024

Cf. A181701, A181595, A181596, A000396.

32128 removed by R. J. Mathar, Dec 05 2010

a(5) from Amiram Eldar, Apr 29 2024

A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).

Original entry on oeis.org

18, 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, 4951760152529835082242850816, 6129982163463555428116476125461573244012649752219877376

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

The associated exponents k are in A000043: 2, 3, 5, 7, 13, 17, 19 ,31, 61, ...

One can prove that, if m = 2^(k-1)*(2^k-1) is a perfect number, then m*2^k and m*(2^k-1) are both in A181595. Thus every even term in A000396 is a difference of two terms in A181595.

			With k=3, m = 2^(k-1)*(2^k - 1) = 2^2*(8 - 1) = 28 is a perfect number (A000396), so m*(2^k - 1) = 28*7 = 196 is in the sequence. - _Michael B. Porter_, Jul 19 2016

Cf. A000043, A000396, A181595, A181596, A181701, A181710.

If odd perfect numbers do not exist, then a(n) = A181710(n) - A000396(n).

a(n) = A019279(n)*(A000668(n))^2 if there are no odd superperfect numbers. - César Aguilera, Jun 13 2017

Definition condensed by R. J. Mathar, Dec 05 2010

cp's OEIS Frontend

A181598 Numbers m with divisor 8 | m and abundance sigma(m)-2*m = 8.

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A181701 Near-perfect numbers (A181595) of the form 2^(t-1)*(2^t-2^k-1), where 2^t-2^k-1 is prime, k>=1, t>k.

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A181597 (N\{4})-perfect numbers, i.e., numbers m for which sigma(m)-4 = 2m, if 4|m, otherwise sigma(m) = 2m.

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

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A181710 Near-perfect numbers (A181595) of the form m*2^p, where m = 2^(p-1)*(2^p-1) is a perfect number (A000396).

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

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A181702 Numbers of the form 2^r*p with p an odd prime, which are in A181595 but not in A181701.

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A181711 Numbers of the form m*(2^k-1), where m = 2^(k-1)*(2^k-1) is a perfect number (A000396).

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A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).

A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).