A068404 -id:A068404 - cp's OEIS frontend

A307114 Primitive 4-abundant numbers: Numbers k such that sigma(k) > 4k (A068404) all of whose proper divisors d are 4-deficient numbers (having sigma(d) < 4d).

27720, 50400, 75600, 85680, 95760, 105840, 115920, 120120, 128520, 141120, 143640, 176400, 180180, 184800, 205920, 207900, 214200, 218400, 235620, 239400, 264600, 289800, 292320, 299880, 308880, 312480, 314160, 351120, 371280, 372960, 414960, 425040, 438480

Offset: 1

Amiram Eldar, Mar 25 2019

nonn

Analogous to A071395 with abundancy index 4 instead of 2.

Paul Erdős and János Surányi, Topics in the Theory of Numbers, New York: Springer, 2003, p. 243.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, Primitive alpha-abundant numbers, Mathematics of Computation, Vol. 43, No. 167 (1984), pp. 263-270.
Paul Erdős, On additive arithmetical functions and applications of probability to number theory, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, Vol. 3 (1956), pp. 13-19.
Paul Erdős, Remarks on number theory. I: On primitive alpha-abundant numbers, Acta Arithmetica., Vol. 5, No. 1 (1959), pp. 25-33, alternative link.

Cf. A000203, A068404, A071395.

Select[Range@500000, DivisorSigma[1, #] > 4 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 4 # &, Most@ Divisors@ #] == 1 &] (* after Michael De Vlieger at A071395 *)

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

Original entry on oeis.org

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

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 *)

A215264 Numbers k such that sigma(k) > 5*k.

Original entry on oeis.org

122522400, 147026880, 183783600, 205405200, 220540320, 232792560, 245044800, 273873600, 294053760, 328648320, 367567200, 410810400, 428828400, 441080640, 465585120, 490089600, 492972480, 497296800, 514594080, 537213600, 547747200, 551350800, 563603040

Offset: 1

Donovan Johnson, Aug 07 2012

nonn

The asymptotic density of this sequence is > 1/a(1) ~ 8*10^(-9). Wall et al. (1972) proved that it is < 0.0122. - Amiram Eldar, Feb 13 2021

			sigma(122522400) = 614210688 and 614210688 > 5 * 122522400.

Donovan Johnson, 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.
Gordon L. Miller and Mary T. Whalen, Multiply Abundant Numbers, School Science and Mathematics, Volume 95, Issue 5 (May 1995), pp. 256-259.
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.

Cf. A000203, A001221, A023199, A068403, A068404.

for(n=122522400, 563603040, if(sigma(n)>5*n, print1(n ", ")))

A001221(a(n)) >= 6 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

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).

A380848 Numbers k such that A380845(k) = 4*k.

Original entry on oeis.org

123832800, 247695840, 268337160, 495421920, 536707080, 990874080, 1073446920, 1981778400, 2146926600, 3963587040, 4293885960, 7927204320, 8587804680, 15854438880, 17175642120, 31708908000, 34351317000, 63417846240, 68702666760, 124884879840, 126713795040, 126835722720

Offset: 1

Amiram Eldar, Feb 05 2025

Analogous to 4-perfect numbers (A027687) with A380845 instead of A000203.
All the terms are 4-abundant numbers (A068404), because A380845(k) <= A000203(k) with equality only when k is a power of 2, and powers of 2 are deficient numbers (A005100).
Are there numbers k such that A380845(k) = m*k for integers m >= 5? There are none below 1.6*10^11.

Cf. A000203, A005100, A027687, A380845, A380846, A380847.

Subsequence of A068404.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] == 4*k]; Select[Range[3*10^8], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 4*k;}

a(19)-a(22) from Jinyuan Wang, Feb 12 2025

A380931 Numbers k such that A380845(k) > 4*k.

Original entry on oeis.org

5155920, 7733880, 10311840, 15467760, 20623680, 30935520, 41247360, 46403280, 61871040, 61901280, 75546240, 82494720, 87693480, 92806560, 103168800, 103194000, 113513400, 123742080, 123802560, 134152200, 140540400, 151092480, 151351200, 162162000, 164989440, 175386960

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to 4-abundant numbers (A068404) with A380845 instead of A000203.

			5155920 is a term since A380845(5155920) = 21067042 > 4 * 5155920 = 20623680.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000203, A380845, A380846, A380847, A380848.
Subsequence of A068404, A380929 and A380931.
Similar sequences: A307114, A340110.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 4*k]; Select[Range[10^8], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 4*k;}

A291458 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 4.

Original entry on oeis.org

27720, 60480, 65520, 90720, 98280, 105840, 115920, 120120, 120960, 128520, 131040, 143640, 151200, 163800, 180180, 191520, 205920, 207900, 211680, 218400, 229320, 235620, 241920, 249480, 264600, 272160, 289800, 292320, 312480, 332640, 360360, 372960, 393120, 414960

Offset: 1

Paolo P. Lava, Aug 24 2017

Case k=2 are the admirable numbers (A111592).

Subset of A023198, A068404, A204831, A230608.

			One of the proper divisors of 27720 is 360 and sigma(27720) - 4*360 = 112320 - 1440 = 110880 = 4*27720.
One of the proper divisors of 115920 is 144 and sigma(115920) - 4*144 = 464256 - 576 = 463680 = 4*115920.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A023198, A068404, A111592, A204831, A230608, A291457, A291459.

with(numtheory): P:=proc(q,h) local a,k,n; for n from 1 to q do a:=sort([op(divisors(n))]);
for k from 1 to nops(a)-1 do if sigma(n)-h*a[k]=h*n then print(n); break; fi; od; od; end: P(10^9,4);

With[{k = 4}, Select[Range[5 * 10^5], Function[n, AnyTrue[Most@ Divisors@ n, DivisorSigma[1, n] - k # == k n &]]]] (* Michael De Vlieger, Aug 24 2017 *)
(* or *)
k=4; Select[Range[5*^5], (t = DivisorSigma[1, #]/k - #; #>t>0 && IntegerQ[t] && Mod[#, t] == 0) &] (* much faster, Giovanni Resta, Aug 25 2017 *)

A335030 Numbers m that are not practical and have an abundancy index sigma(m)/m which is larger than that of any smaller number that is not practical.

Original entry on oeis.org

3, 9, 10, 44, 70, 102, 350, 372, 1608, 3492, 6096, 10380, 44040, 100260, 180240, 425160, 1744560, 2425080, 5509980, 10048080, 23614920, 97639920, 396315360, 900229680, 2519017200, 3113704440, 12870562320, 52307529120

Offset: 1

Amiram Eldar, May 20 2020

None of the terms are superabundant (A004394) since all the superabundant numbers are practical numbers (A005153).
The least term m that is k-abundant (having sigma(m)/m > k) for k = 2, 3, ... is A005101(14) = 70, A068403(896) = 44040, A068404(792087) = 3113704440, ...
What is the least 5-abundant number (A215264) that is not practical?

			The first 5 numbers that are not practical are m = 3, 5, 7, 9, 10. Their abundancy indices sigma(m)/m are 1.333..., 1.2, 1.142..., 1.444..., 1.8. The record values occur at 3, 9 and 10.

Cf. A000203 (sigma), A004394, A005153, A237287, A330244, A335029.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; seq = {}; rm = 1; Do[fct = FactorInteger[n]; r = Times@@((First/@fct^ (1+Last/@ fct)-1)/(First/@fct-1))/n; If[r > rm && !pracQ[fct], rm = r; AppendTo[seq, n]], {n, 3, 10^5}]; seq

A340110 Coreful 4-abundant numbers: numbers k such that csigma(k) > 4*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

10584000, 12700800, 15876000, 19051200, 21168000, 22226400, 25401600, 29635200, 31752000, 37044000, 38102400, 42336000, 44452800, 47628000, 50803200, 52920000, 55566000, 57153600, 59270400, 63504000, 64033200, 66679200, 74088000, 76204800, 79380000, 84672000

Offset: 1

Amiram Eldar, Dec 28 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

Analogous to A068404 as A308053 is analogous to A005101.

			10584000 is a term since csigma(10584000) = 42653520 > 4 * 10584000.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A308053 and A340109.
Cf. A007947, A057723.
Similar sequences: A068404, A307114.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^8], s[#] > 4*# &]

cp's OEIS Frontend

A307114 Primitive 4-abundant numbers: Numbers k such that sigma(k) > 4k (A068404) all of whose proper divisors d are 4-deficient numbers (having sigma(d) < 4d).

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

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

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A215264 Numbers k such that sigma(k) > 5*k.

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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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A380848 Numbers k such that A380845(k) = 4*k.

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A380931 Numbers k such that A380845(k) > 4*k.

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A291458 Numbers n having a proper divisor d such that sigma(n) - k*d = k*n. Case k = 4.

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A291458 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 4.