A119240 -id:A119240 - cp's OEIS frontend

A068403 Numbers k such that sigma(k) > 3*k.

180, 240, 360, 420, 480, 504, 540, 600, 660, 720, 780, 840, 900, 960, 1008, 1080, 1200, 1260, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1872, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640

Offset: 1

Benoit Cloitre, Mar 02 2002

Davenport shows that these numbers have positive density. Are there good bounds for the density?
G. Miller & M. Whalen suggested that 1018976683725 (3^3*5^2*7^2*11*13*17*19*23*29) might be the smallest odd number in the sequence (a fact now, see A119240 and A023197). - Michel Marcus, May 01 2013
From Amiram Eldar, Feb 13 2021: (Start)
Behrend (1933) found the bounds (0.009, 0.110) for the asymptotic density.
Wall et al. (1972) found the bounds (0.0186, 0.0461).
The upper bound was reduced to 0.0214614 using Deléglise's method by McDaniel College (2010). (End)
Note that 1018976683725, the smallest odd term in this sequence, is A053624(51). - Charles R Greathouse IV, Jan 09 2025

Harold Davenport, Über numeri abundantes, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl., No. 6 (1933), pp. 830-837.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Felix Behrend, Über numeri abundantes II, Preuss. Akad. Wiss. Sitzungsber., Vol. 6 (1933), pp. 280-293; alternative link.
Marc Deléglise, Bounds for the Density of Abundant Integers, Experimental Mathematics, Vol. 7, No. 2 (1998), pp. 137-143.
Richard Laatsch, Measuring the Abundancy of Integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92, alternative link.
Gordon L. Miller and Mary T. Whalen, Multiply Abundant Numbers, School Science and Mathematics, Volume 95, Issue 5 (May 1995), pp. 256-259.
Summer 2010 research group on Abundancy, Abundancy Bounds 2010, McDaniel College, 2010.
Charles R. Wall, Phillip L. Crews and Donald B. Johnson, Density Bounds for the Sum of Divisors Function, Mathematics of Computation, Vol. 26, No. 119 (1972), pp. 773-777; Errata, Vol. 31, No. 138 (1977), p. 616.

Terms not divisible by 6 are in A126104.
Cf. A001221, A068404, A215264.
Cf. A005820 (3-perfect numbers).

A068403:=n->`if`((numtheory)[sigma](n) > 3*n, n, NULL): seq(A068403(n), n=1..5*10^3); # Wesley Ivan Hurt, Apr 09 2017

Select[Range[180, 2000], 3*# < Plus@@Divisors[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
Select[Range[3000],DivisorSigma[1,#]>3#&] (* Harvey P. Dale, Aug 12 2023 *)

for(n=1, 3000, if(sigma(n)>3*n, print1(n,", "))) \\ Indranil Ghosh, Apr 10 2017

from sympy import divisor_sigma
print([n for n in range(180, 3001) if divisor_sigma(n)>3*n]) # Indranil Ghosh, Apr 10 2017

A001221(a(n)) >= 3 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

a(n) ~ k*n for some constant k with 46 < k < 54. - Charles R Greathouse IV, Jan 21 2025

A023197 Numbers k such that sigma(k) >= 3*k.

Original entry on oeis.org

120, 180, 240, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1080, 1200, 1260, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1872, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520

Offset: 1

David W. Wilson

nonn

Sometimes called 3-abundant numbers (but compare the comments in A033880). The first odd number is A119240(3) = 1018976683725. - T. D. Noe, Mar 31 2011

Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, 2000, p 260.

T. D. Noe, Table of n, a(n) for n = 1..10000
Richard Laatsch, Measuring the Abundancy of Integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92, alternative link.

Cf. A000203, A001221, A119240.

See A033880 for definition of k-abundancy.

select(t -> numtheory:-sigma(t) >= 3*t, [$1..10000]); # Robert Israel, Dec 28 2014

Select[Range[10000], DivisorSigma[1,#] >= 3*#&] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)

A001221(a(n)) >= 3 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

A096536 Numbers k such that k, k+1, k+2 are all abundant.

Original entry on oeis.org

171078830, 268005374, 321893648, 336038624, 487389824, 600350750, 663249950, 668645054, 938109248, 1053424448, 1079741024, 1102433408, 1139364224, 1148927624, 1267293950, 1275861950, 1310259950, 1344330350, 1352253824

Offset: 1

John L. Drost, Aug 13 2004

nonn

The entries shown are all even, the first odd k would have to have sigma(k*(k+2)) > 4k*(k+2) so k > 10^19 (cf. A119240).
From Amiram Eldar, Oct 02 2022: (Start)
The least term that is == 1 (mod 3) is a(1292) = 55959128224, and the least term that is divisible by 3 is a(1590) = 68972878974.
The numbers of terms not exceeding 10^k, for k = 9, 10, ..., are 9, 226, 2298, 22583, ... . Apparently, the asymptotic density of this sequence exists and equals 2.2...*10^(-8). (End)

			For 171078830 = 2*5*13*23*29*1973, sigma(n)/n = 2.09355, for 171078831 = 3^3*7*11*19*61*71, sigma(n)/n = 2.00396 and for 171078832 = 2^4*31*344917, sigma(n)/n = 2.00000579.

Amiram Eldar, Table of n, a(n) for n = 1..22583 (terms below 10^12; terms 1..1000 from Donovan Johnson)
Carlos Rivera, Puzzle 878. Consecutive abundant integers, The Prime Puzzles & Problems Connection.
Carlos Rivera, Puzzle 880. Consecutive odd abundant integers, The Prime Puzzles & Problems Connection.

Subsequence of A005101 and A096399.

Cf. A119240.

isab(x) = sigma(x) > 2*x; \\ A005101
isok(k) = isab(k) && isab(k+1) && isab(k+2); \\ Michel Marcus, Nov 19 2022

a(15)-a(19) from Donovan Johnson, Dec 29 2008

A023198 Numbers k such that sigma(k) >= 4*k.

Original entry on oeis.org

27720, 30240, 32760, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 141120, 143640, 151200, 163800, 166320, 171360, 176400, 180180, 181440, 184800

Offset: 1

David W. Wilson

nonn

Called 4-abundant numbers. The first odd number is A119240(4) = 1853070540093840001956842537745897243375. - T. D. Noe, Mar 31 2011

Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, 2000, p 260.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Union of A027687 and A068404.

Select[Range[1000000], DivisorSigma[1, #] >= 4*#&] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)

A358412 Least number k coprime to 2 and 3 such that sigma(k)/k >= n.

Original entry on oeis.org

1, 5391411025, 5164037398437051798923642083026622326955987448536772329145127064375

Offset: 1

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 5^2*7*...*29 and a(3) = 5^4*7^3*11^2*13^2*17*...*157 ~ 5.16404*10^66. a(4) = 5^5*7^4*11^3*13^3*17^2*19^2*23^2*29^2*31^2*37^2*41*...*853 ~ 1.83947*10^370 is too large to display.

			a(2) = A047802(2) = 5391411025 is the smallest abundant number coprime to 2 and 3.
Even if there is a number k coprime to 2 and 3 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=3..m+2} (prime(i)/(prime(i)-1)) => m >= 33, and we have k >= prime(3)^2*...*prime(35)^2 ~ 6.18502*10^112 > A358413(2) ~ 5.16403*10^66. So a(3) = A358413(2).
Even if there is a number k coprime to 2 and 3 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=3..m+2} (prime(i)/(prime(i)-1)) => m >= 140, and we have k >= prime(3)^2*...*prime(140)^2*prime(141)*prime(142) ~ 2.65585*10^669 > A358414(2) ~ 1.83947*10^370. So a(4) = A358414(2).

Jianing Song, Table of n, a(n) for n = 1..4
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), this sequence (p=5), A358418 (p=7), A358419 (p=11).

A358413 Smallest 3-abundant number (sigma(x) > 3x) which is not divisible by any of the first n primes.

Original entry on oeis.org

180, 1018976683725, 5164037398437051798923642083026622326955987448536772329145127064375

Offset: 0

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(0) = 2^2*3^2*5, a(1) = 3^3*5^2*7^2*11*13*17*19*23*29, and a(2) = 5^4*7^3*11^2*13^2*17*...*157 ~ 5.16404*10^66. a(3) = 7^3*11^3*13^2*17^2*19^2*23^2*29^2*31*...*569 ~ 2.54562*10^239 and a(4) = 11^3*13^3*17^2*...*47^2*53*...*1597 ~ 3.99515*10^688 are too large to display.

			a(1) = A119240(3) = 1018976683725 is the smallest 3-abundant odd number.
a(2) = A358412(3) = 5164037398437051798923642083026622326955987448536772329145127064375 is the smallest 3-abundant number that is coprime to 2 and 3.

Jianing Song, Table of n, a(n) for n = 0..4
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Cf. A068403 (3-abundant numbers).
Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), this sequence (k=3), A358414 (k=4).
Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), A358419 (p=11).

A119239 Oddly superabundant numbers: odd n with sigma(n)/n > sigma(k)/k for all odd k < n.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 135135, 225225, 405405, 675675, 2027025, 2297295, 3828825, 6891885, 11486475, 34459425, 43648605, 72747675, 130945815, 218243025, 654729075, 1003917915, 1527701175

Offset: 1

T. D. Noe, May 09 2006

nonn

Every oddly colossally abundant number (A110464) is in this sequence.

a(8) = 945 is the first term with abundancy > 2, a(41) = 1018976683725 is the first term with abundancy > 3, and a(141) = 1853070540093840001956842537745897243375 is the first term with abundancy > 4. See A119240. - Antti Karttunen, Jul 21 2025

T. D. Noe, Table of n, a(n) for n = 1..1000
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Walter Nissen, Abundancy : Some Resources

Cf. A004394 (superabundant numbers), A005231 (odd abundant numbers), A053624 (highly composite odd numbers), A119240.

Cf. also A171929, A228059, A386423.

rec=0; lst={}; Do[abun=DivisorSigma[1,n]/n; If[abun>rec, rec=abun; AppendTo[lst,n]], {n,1,10^6,2}]; lst

r=0;forstep(n=1,1e6,2,t=sigma(n)/n;if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Nov 27 2013

Definition clarified by Jonathan Sondow, Dec 08 2011

A358414 Smallest 4-abundant number (sigma(x) > 4x) which is not divisible by any of the first n primes.

Original entry on oeis.org

27720, 1853070540093840001956842537745897243375

Offset: 0

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(0) = 2^3*3^2*5*7*11 and a(1) = 3^5*5^3*7^2*11^2*13*...*89 ~ 1.85307*10^39. a(2) = 5^5*7^4*11^3*13^3*17^2*19^2*23^2*29^2*31^2*37^2*41*...*853 ~ 1.83947*10^370, a(3) = 7^5*11^3*13^3*17^3*19^3*23^2*...*97^2*101*...*4561 ~ 1.11116*10^1986, and a(4) = 11^4*13^4*17^3*19^3*23^3*29^3*31^3*37^2*...*181^2*191*...*18493 ~ 2.99931*10^8063 are too large to display.

			a(1) = A119240(4) = 1853070540093840001956842537745897243375 is the smallest 4-abundant odd number.
a(2) = A358412(4) ~ 1.83947*10^370 is the smallest 4-abundant number that is coprime to 2 and 3.

Jianing Song, Table of n, a(n) for n = 0..2
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Cf. A068404 (4-abundant numbers).
Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), this sequence (k=4).
Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), A358419 (p=11).

A358418 Least number k coprime to 2, 3, and 5 such that sigma(k)/k >= n.

Original entry on oeis.org

1, 20169691981106018776756331

Offset: 1

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 7^2*11^2*13*...*67 ~ 2.01697*10^25. a(3) = 7^3*11^3*13^2*17^2*19^2*23^2*29^2*31*...*569 ~ 2.54562*10^239 and a(4) = 7^5*11^3*13^3*17^3*19^3*23^2*...*97^2*101*...*4561 ~ 1.11116*10^1986 are too large to display.

			a(2) = A047802(3) = 20169691981106018776756331 is the smallest abundant number coprime to 2, 3, and 5.
Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 97, and we have k >= prime(4)^2*...*prime(100)^2 ~ 2.46692*10^436 > A358413(3) ~ 2.54562*10^239. So a(3) = A358413(3).
Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 606, and we have k >= prime(4)^2*...*prime(607)^2*prime(608)*prime(609) ~ 6.54355*10^3814 > A358414(3) ~ 1.11116*10^1986. So a(4) = A358414(3).

Jianing Song, Table of n, a(n) for n = 1..3
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), this sequence (p=7), A358419 (p=11).

A358419 Least number k coprime to 2, 3, 5, and 7 such that sigma(k)/k >= n.

Original entry on oeis.org

1, 49061132957714428902152118459264865645885092682687973

Offset: 1

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 11^2*13^2*17*...137 ~ 4.90611*10^52. a(3) = 11^3*13^3*17^2*...*47^2*53*...*1597 ~ 3.99515*10^688 and a(4) = 11^4*13^4*17^3*19^3*23^3*29^3*31^3*37^2*...*181^2*191*...*18493 ~ 2.99931*10^8063 are too large to display.

			a(2) = A047802(4) = 49061132957714428902152118459264865645885092682687973 is the smallest abundant number coprime to 2, 3, 5, and 7.
Even if there is a number k coprime to 2, 3, 5, and 7 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=5..m+4} (prime(i)/(prime(i)-1)) => m >= 240, and we have k >= prime(5)^2*...*prime(244)^2 ~ 1.60834*10^1297 > A358413(4) ~ 3.99515*10^688. So a(3) = A358413(4).
Even if there is a number k coprime to 2, 3, 5, and 7 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=5..m+4} (prime(i)/(prime(i)-1)) => m >= 2096, and we have k >= prime(5)^2*...*prime(2098)^2*prime(2099)*prime(2100) ~ 6.21439*10^15801 > A358414(4) ~ 2.99931*10^8063. So a(4) = A358414(4).

Jianing Song, Table of n, a(n) for n = 1..3
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), this sequence (p=11).

cp's OEIS Frontend

A068403 Numbers k such that sigma(k) > 3*k.

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A023197 Numbers k such that sigma(k) >= 3*k.

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A096536 Numbers k such that k, k+1, k+2 are all abundant.

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A023198 Numbers k such that sigma(k) >= 4*k.

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A358412 Least number k coprime to 2 and 3 such that sigma(k)/k >= n.

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A358413 Smallest 3-abundant number (sigma(x) > 3x) which is not divisible by any of the first n primes.

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A119239 Oddly superabundant numbers: odd n with sigma(n)/n > sigma(k)/k for all odd k < n.

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A358414 Smallest 4-abundant number (sigma(x) > 4x) which is not divisible by any of the first n primes.

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