A322287 -id:A322287 - cp's OEIS frontend

A173490 Even abundant numbers (even numbers n whose sum of divisors exceeds 2n).

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240

Offset: 1

Daniel Forgues, Nov 22 2010

Set difference of abundant numbers A005101 by odd abundant numbers A005231.
While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!  Thus the first 231 terms of this sequence are the same as for sequence A005101 of abundant numbers.
Dickson proves that, for each m and n, there are only a finite number of these numbers having a factor 2^m and n distinct odd prime factors. - T. D. Noe, Mar 31 2011
The asymptotic density of this sequence is in the interval (0.245548, 0.245578) (based on the known bounds on the densities of A005101 and A005231; see A302991 and A322287). - Amiram Eldar, Mar 11 2024

T. D. Noe, Table of n, a(n) for n = 1..10000
L. E. Dickson, Even abundant numbers, American Journal of Mathematics 35 (1913), pp. 423-426.

Cf. A005101, A005231, A039725, A302991, A322287.

Select[2*Range[150], DivisorSigma[1, #] > 2 # &] (* T. D. Noe, Jun 25 2012 *)

is(n)=n%2==0 && sigma(n,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

a(n) = 2 * A039725(n). - Amiram Eldar, Mar 11 2024

A039725 Even abundant numbers divided by 2.

Original entry on oeis.org

6, 9, 10, 12, 15, 18, 20, 21, 24, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 48, 50, 51, 52, 54, 56, 57, 60, 63, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 88, 90, 93, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 117, 120, 123, 126, 129, 130, 132, 135, 136

Offset: 1

N. J. A. Sloane. This was included in the 1973 "Handbook", but was then dropped from the database. Resubmitted by James Sellers. Entry revised by N. J. A. Sloane, Jun 12 2012

While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!

The asymptotic density of this sequence is in the interval (0.491096, 0.491156) (based on the known bounds on the densities of A005101 and A005231; see A302991 and A322287). - Amiram Eldar, Mar 11 2024

			The first even abundant number is 12, so 12/2 = 6 is the first element in this sequence.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A005101, A005231, A173490, A302991, A322287.

Select[Range[2,300,2],DivisorSigma[1,#]>2*#&]/2 (* Harvey P. Dale, Mar 22 2020 *)

a(n) = A173490(n) / 2.

Corrected and edited by Daniel Forgues, Nov 22 2010

A343794 Numbers k > 0 such that 630*k + 315 is not an abundant number (A005101).

Original entry on oeis.org

53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158, 165, 168, 173, 174, 176, 179, 183, 186, 189, 191, 194, 198, 200, 204, 209, 210, 215

Offset: 1

Amiram Eldar, Apr 29 2021

nonn

630*k + 315 is an abundant number for the first 52 positive values of k.
The number of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 19, 276, 2242, 22249, 235300, 2319944, 22958712, 230566888, 2308563768, 23063629594, ... Apparently the asymptotic density of this sequence is 0.230...
There are 2048662 odd abundant numbers (A005231) below 10^9, of them 1213732 are of the form 630*k + 315. Apparently, the asymptotic density of abundant numbers of this form within the odd abundant numbers is about 0.6.
From Jianing Song, May 30 2022: (Start)
Numbers k > 0 such that (2*k+1)/sigma(2*k+1) <= 105/104.
Contains (p^i-1)/2 for all primes p >= 107 and i >= 1.
Since 315*p is abundant for primes p = 2, 3, 5, 7, 11, ..., 103, the prime factors of 2*k+1 are at least 107 if k is a term of this sequence. Hence we have a(n) = A005097(n+26) = (prime(n+27)-1)/2 for n <= 1354, whereas 2*a(1355)+1 = 11449 = 107^2.
The smallest term k such that 2*k+1 is not a prime power is k = a(4872), with 2*k+1 = 211*223. (End)

			53 is a term since 630*53 + 315 = 33705 is not an abundant number.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd ed., Penguin, 1997, p. 155.
M. T. Whalen and C. L. Miller, Odd abundant numbers: some interesting observations, Journal of Recreational Mathematics 22 (1990), pp. 257-261.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 113.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Vol. 37, No. 2 (2005), pp. 73-75.

Cf. A005101, A005231, A263837, A322287.

abQ[n_] := DivisorSigma[1, n] > 2*n; Select[Range[200], !abQ[630*# + 315] &]

cp's OEIS Frontend

A173490 Even abundant numbers (even numbers n whose sum of divisors exceeds 2n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A039725 Even abundant numbers divided by 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A343794 Numbers k > 0 such that 630*k + 315 is not an abundant number (A005101).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica