A153501 -id:A153501 - cp's OEIS frontend

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

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.

A005820 3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

Original entry on oeis.org

120, 672, 523776, 459818240, 1476304896, 51001180160

Offset: 1

N. J. A. Sloane

These six terms are believed to comprise all 3-perfect numbers. - cf. the MathWorld link. - Daniel Forgues, May 11 2010
If there exists an odd perfect number m (a famous open problem) then 2m would be 3-perfect, since sigma(2m) = sigma(2)*sigma(m) = 3*2m. - Jens Kruse Andersen, Jul 30 2014
According to the previous comment from Jens Kruse Andersen, proving that this sequence is complete would imply that there are no odd perfect numbers. - Farideh Firoozbakht, Sep 09 2014
If 2 were prepended to this sequence, then it would be the sequence of integers k such that numerator(sigma(k)/k) = A017665(k) = 3. - Michel Marcus, Nov 22 2015
From  Antti Karttunen, Mar 20 2021, Sep 18 2021, (Start):
Obviously, any odd triperfect numbers k, if they exist, have to be squares for the condition sigma(k) = 3*k to hold, as sigma(k) is odd only for k square or twice a square. The square root would then need to be a term of A097023, because in that case sigma(2*k) = 9*k. (See illustration in A347391).
Conversely to Jens Kruse Andersen's comment above, any 3-perfect number of the form 4k+2 would be twice an odd perfect number. See comment in A347870.
(End)

			120 = 2^3*3*5;  sigma(120) = (2^4-1)/1*(3^2-1)/2*(5^2-1)/4 = (15)*(4)*(6) = (3*5)*(2^2)*(2*3) = 2^3*3^2*5 = (3) * (2^3*3*5) = 3 * 120. - _Daniel Forgues_, May 09 2010

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 120, p. 42, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, B2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Stewart, L'univers des nombres, "Les nombres multiparfaits", Chap.15, pp 82-5, Belin/Pour la Science, Paris 2000.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 142.
David Wells, "The Penguin Book of Curious and Interesting Numbers," Penguin Books, London, 1986, pages 135, 159 and 185.

Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Kevin A. Broughan and Qizhi Zhou, Divisibility by 3 of even multiperfect numbers of abundancy 3 and 4, JIS 13 (2010) 10.1.5
Alfred Brousseau, Number Theory Tables, Fibonacci Association, San Jose, CA, 1973, p. 138.
Seth Colbert-Pollack, Judy Holdener, Emily Rachfal, and Yanqi Xu, A DIY Project: Construct Your Own Multiply Perfect Number!, Math Horizons, Vol. 28, pp. 20-23, February 2021.
Farideh Firoozbakht and Maximilian F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Achim Flammenkamp, The Multiply Perfect Numbers Page [This page contains a lot of useful information, but be careful, not all the statements are correct. For example, it appears to claim that the six terms of this sequence are known to be complete, which is not the case. - _N. J. A. Sloane_, Sep 10 2014]
James Grime and Brady Haran, The Six Triperfect Numbers, Numberphile video (2018).
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
Fred Helenius, Link to Glossary and Lists
Masao Kishore, Odd Triperfect Numbers, Mathematics of Computation, vol. 42, no. 165, 1984, pp. 231-233.
Gérard P. Michon, Multiperfect and hemiperfect numbers
Walter Nissen, Abundancy : Some Resources
N. J. A. Sloane & A. L. Brown, Correspondence, 1974
Eric Weisstein's World of Mathematics, Multiperfect Number
Eric Weisstein's World of Mathematics, Sous-Double
Wikipedia, Multiply perfect number, (section Triperfect numbers)

Cf. A000203, A000396, A007539, A017665, A019278, A027687, A046060, A046061, A068403, A075701, A097023, A171266, A259302, A259303, A306373, A326051, A326181, A329189, A335141, A335254, A347383, A347391.
Subsequence of the following sequences: A007691, A069085, A153501, A216780, A292365, A336458, A336461, A336745, and if there are no odd terms, then also of A334410.
Positions of 120's in A094759, 119's in A326200.

A005820:=n->`if`(numtheory[sigma](n) = 3*n, n, NULL): seq(A005820(n), n=1..6*10^5); # Wesley Ivan Hurt, Oct 15 2017

triPerfectQ[n_] := DivisorSigma[1, n] == 3n; A005820 = {}; Do[If[triPerfectQ[n], AppendTo[A005820, n]], {n, 10^6}]; A005820 (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Select[Range[10^6],DivisorSigma[1,#]==3#&] (* Harvey P. Dale, Jul 03 2023 *)

isok(n) = sigma(n, -1) == 3; \\ Michel Marcus, Nov 22 2015

a(n) = 2*A326051(n). [provided no odd triperfect numbers exist] - Antti Karttunen, Jun 13 2019

Wells gives the 6th term as 31001180160, but this is an error.

Edited by Farideh Firoozbakht and N. J. A. Sloane, Sep 09 2014 to remove some incorrect statements.

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

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

Original entry on oeis.org

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

A271816 Deficient-perfect numbers: Deficient numbers n such that n/(2n-sigma(n)) is an integer.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 44, 64, 128, 136, 152, 184, 256, 512, 752, 884, 1024, 2048, 2144, 2272, 2528, 4096, 8192, 8384, 12224, 16384, 17176, 18632, 18904, 32768, 32896, 33664, 34688, 49024, 63248, 65536, 85936, 106928, 116624, 117808, 131072, 262144, 524288, 526688, 527872, 531968, 556544, 589312, 599072, 654848, 709784

Offset: 1

Carlo Francisco E. Adajar, Apr 14 2016

nonn

Every power of 2 is part of this sequence, with 2n - sigma(n) = 1.
Any odd element of this sequence must be a perfect square with at least four distinct prime factors (Tang and Feng). The smallest odd element of this sequence is a(74) = 9018009 = 3^2 * 7^2 * 11^2 * 13^2, with 2n - sigma(n) = 819.
a(17) = 884 = 2^2 * 13 * 17 is the smallest element with three distinct prime factors.
For all n > 1 in this sequence, 5/3 <= sigma(n)/n < 2. - Charles R Greathouse IV, Apr 15 2016

			When n = 1, 2, 4, 8, 2n - sigma(n) = 1.
When n = 10, sigma(10) = 18 and so 2*10 - 18 = 2, which divides 10.

Giovanni Resta, Table of n, a(n) for n = 1..273 (terms < 2*10^12)
BYU Computational Number Theory Group, Odd, spoof perfect factorizations, arXiv:2006.10697 [math.NT], 2020.
Jose A. B. Dris, Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, No. 2 (2017), pp. 12-20, arXiv preprint, arXiv:1610.01868 [math.NT], 2016.
Jose Arnaldo Bebita Dris and Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
Hùng Việt Chu, Divisibility of Divisor Functions of Even Perfect Numbers, J. Int. Seq., Vol. 24 (2021), Article 21.3.4.
Cui-Fang Sun and Zhao-Cheng He, On odd deficient-perfect numbers with four distinct prime divisors, arXiv:1908.04932 [math.NT], 2019.
Judy Holdener and Emily Rachfal, Perfect and Deficient Perfect Numbers, The American Mathematical Monthly, Vol. 126, No. 6 (2019), pp. 541-546.
M. Tang, X. Z. Ren, and M. Li, On Near-Perfect and Deficient-Perfect Numbers, Colloq. Math. 133 (2013), 221-226.
M. Tang and M. Feng, On Deficient-Perfect Numbers, Bull. Aust. Math. Soc. 90 (2014), 186-194.

Cf. A000203, A005100, A033879.
Deficient analog of A153501. Setwise difference A097498\A153501.
Contains A000079.

q:= k-> (s-> s>0 and irem(k, s)=0)(2*k-numtheory[sigma](k)):
select(q, [$1..500000])[];  # Alois P. Heinz, Aug 26 2023

ok[n_] := Block[{d = DivisorSigma[1, n]}, d < 2*n && Divisible[n, 2*n - d]]; Select[Range[10^5], ok] (* Giovanni Resta, Apr 14 2016 *)

isok(n) = ((ab = (sigma(n)-2*n))<0) && (n % ab == 0); \\ Michel Marcus, Apr 15 2016

2^k is always an element of this sequence.

If 2^(k+1) + 2^t - 1 is an odd prime and t <= k, then n = 2^k(2^(k+1) + 2^t - 1) is deficient-perfect with 2n - sigma(n) = 2^t. In fact, these are the only terms with two distinct prime factors. (Tang et al.)

A181596 Abundance of A181595(n).

Original entry on oeis.org

4, 3, 2, 12, 10, 8, 4, 2, 7, 56, 78, 8, 2, 2, 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, 32, 2272, 32, 32, 127, 16256, 32, 32, 4, 536, 8, 32, 8, 52, 16, 32, 41044, 64, 512, 128, 64, 16, 4, 2, 8

Offset: 1

Vladimir Shevelev, Nov 01 2010

nonn

a(n) is a proper divisor of A181595(n).

			Since A181595(1)=12, a(1)=sigma(12)-2*12=28-24=4.

Michel Marcus, Table of n, a(n) for n = 1..200
Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046. arXiv:1011.6160

Cf. A181595, A000396, A005101, A153501, A005820.

Reap[For[n = 12, n <= 2 10^7, n++, abn = DivisorSigma[1, n] - 2n; If[1 < abn < n && Divisible[n, abn], Print[{n, abn}]; Sow[abn]]]][[2, 1]] (* Jean-François Alcover, Oct 28 2018 *)

a(n) = A033880(A181595(n)).

a(10)-a(11) corrected by Vladimir Shevelev, Nov 03 2010
Entries checked, definition shortened by R. J. Mathar, Nov 17 2010
More terms from Michel Marcus, Feb 06 2016

A097498 Numbers k divisible by their abundance sigma(k) - 2*k.

Original entry on oeis.org

1, 2, 4, 8, 10, 12, 16, 18, 20, 24, 32, 40, 44, 56, 64, 88, 104, 120, 128, 136, 152, 184, 196, 224, 234, 256, 368, 464, 512, 650, 672, 752, 884, 992, 1024, 1504, 1888, 1952, 2048, 2144, 2272, 2528, 3724, 4096, 5624, 8192, 8384, 9112, 11096, 12224, 13736

Offset: 1

Reinhard Zumkeller, Aug 24 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..417 (terms 1..348 from Charles R Greathouse IV)
Eric Weisstein's World of Mathematics, Abundance.

Cf. A000079 (subsequence), A033879, A033880.
Disjoint union of A153501 and A271816.
Cf. also A379236.

Select[Range[15000],Divisible[#,DivisorSigma[1,#]-2#]&]//Quiet (* Harvey P. Dale, Mar 15 2018 *)

is(n)=my(t=sigma(n)-2*n); t && n%t==0 \\ Charles R Greathouse IV, Dec 12 2014

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

A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].

Original entry on oeis.org

10, 12, 18, 20, 24, 40, 56, 88, 104, 116, 136, 184, 196, 224, 312, 368, 428, 464, 520, 528, 650, 672, 760, 884, 992, 1472, 1504, 1888, 1952, 2528, 3424, 3724, 4832, 5312, 6464, 7136, 9112, 11096, 11288, 11744, 13216, 15352, 15376, 15872, 15968, 16256, 17816, 17964, 22616, 24448, 26728, 28544, 29296, 30592, 30656

Offset: 1

Antti Karttunen, Jan 05 2025

nonn

Among the first 484 terms, there are no odd numbers, the only squares are 196, 15376, 1032256, and 18 is the only twice square.

			196 is a term as sigma(196) = 399, 2*196 XOR 399 = 7 is not zero, and A048720(7, 89) = 399.

Cf. A000203, A000396, A003987, A048720, A280500, A318467.
Cf. A379234 (subsequence).
Cf. also A097498 (= A153501 U A271816).

divides_in_GF2X(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); !lift(Pa % Pb); };
is_A379236(n) = { my(s=sigma(n), x=bitxor(2*n, s)); (x && divides_in_GF2X(n, x)); };

{k such that k = A048720(A318467(k), x) for some x > 0}.

{k not in A000396 such that A280500(k, A318467(k)) > 0}.

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

cp's OEIS Frontend

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

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A005820 3-perfect (triply perfect, tri-perfect, triperfect or sous-double) numbers: numbers such that the sum of the divisors of n is 3n.

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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.

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

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A271816 Deficient-perfect numbers: Deficient numbers n such that n/(2n-sigma(n)) is an integer.

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A181596 Abundance of A181595(n).

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A097498 Numbers k divisible by their abundance sigma(k) - 2*k.

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