A097498 -id:A097498 - cp's OEIS frontend

A153501 Abundant numbers n such that n/(sigma(n)-2n) is an integer.

12, 18, 20, 24, 40, 56, 88, 104, 120, 196, 224, 234, 368, 464, 650, 672, 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

Offset: 1

Donovan Johnson, Jan 02 2009

nonn

Sigma(n)-2n is the abundance of n.
The only odd term in this sequence < 2*10^12 is 173369889. - Donovan Johnson, Feb 15 2012
Equivalently, the abundancy of n, ab=sigma(n)/n, satisfies the following relation: numerator(ab) = 2*denominator(ab)+1, that is, ab=(2k+1)/k where k is the integer ratio mentioned in definition. - Michel Marcus, Nov 07 2014
The tri-perfect numbers (A005820) are in this sequence, since their abundancy is 3n/n = 3 = (2k+1)/k with k=1. - Michel Marcus, Nov 07 2014

			The abundance of 174592 = sigma(174592)-2*174592 = 43648. 174592/43648 = 4.

Donovan Johnson, Table of n, a(n) for n = 1..200

Intersection of A097498 and A005101.
Disjoint union of A181595 and A005820.
Cf. A000203, A033880.

filter:= proc(n) local s; s:= numtheory:-sigma(n); (s > 2*n) and (n mod (s-2*n) = 0) end proc:
select(filter, [$1..10^5]); # Robert Israel, Nov 07 2014

filterQ[n_] := Module[{s = DivisorSigma[1, n]}, s > 2n && Mod[n, s - 2n] == 0];
Select[Range[10^6], filterQ] (* Jean-François Alcover, Feb 01 2023, after Robert Israel *)

isok(n) = ((ab = (sigma(n)-2*n))>0) && (n % ab == 0) \\ Michel Marcus, Jul 16 2013

def A153501_list(len):
    def is_A153501(n):
        t = sigma(n,1) - 2*n
        return t > 0 and t.divides(n)
    return filter(is_A153501, range(1,len))
A153501_list(1000) # Peter Luschny, Nov 07 2014

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.)

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

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

cp's OEIS Frontend

A153501 Abundant numbers n such that n/(sigma(n)-2n) is an integer.

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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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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].

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

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

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