A215264 -id:A215264 - cp's OEIS frontend

A307115 Primitive 5-abundant numbers: Numbers k such that sigma(k) > 5k (A215264) all of whose proper divisors d are 5-deficient numbers (having sigma(d) < 5d).

122522400, 147026880, 183783600, 205405200, 220540320, 232792560, 273873600, 328648320, 428828400, 492972480, 497296800, 514594080, 537213600, 563603040, 575134560, 581981400, 605404800, 627026400, 629909280, 670269600, 684684000, 710629920, 739458720, 745945200

Offset: 1

Amiram Eldar, Mar 25 2019

nonn

Analogous to A071395 with abundancy index 5 instead of 2.

omega(a(n)) = A001221(a(n)) >= 6. - David A. Corneth, Mar 26 2019

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..1000
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, A001221, A071395, A215264.

Select[Range@500000000, DivisorSigma[1, #] > 5 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 5 # &, 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

A068404 Numbers k such that sigma(k) > 4*k.

Original entry on oeis.org

27720, 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, 191520

Offset: 1

Benoit Cloitre, Mar 02 2002

nonn

This sequence is of positive density, see for example Davenport. The density is between 0.000176 and 0.004521 according to the McDaniel College link. - Charles R Greathouse IV, Sep 07 2012
From Amiram Eldar, Feb 13 2021: (Start)
Behrend (1933) found the bounds (0.00003, 0.025) for the asymptotic density.
Wall et al. (1972) found the bounds (0.0001, 0.0147).
Using Deléglise's method the upper bound for the density found by McDaniel College is 0.000679406. (End)

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.

Cf. A068403, A001221, A215264.

Cf. A027687 (4-perfect numbers).

Select[Range[27720,9!,60], 4*#Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)

A001221(a(n)) >= 4 (Laatsch, 1986). - Amiram Eldar, Nov 07 2020

A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

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, 191520, 194040

Offset: 1

Jaroslav Krizek, Nov 29 2013

nonn

A subsequence of A023198 (numbers with abundancy >= 4). It differs from A023198 from a(31093) on: The term A023198(31093) = 122522400 = A023199(5) = A215264(1) is not in this sequence. It excludes all terms of A215264, but also the 5-perfect numbers A046060, which are neither in this sequence nor in A215264. [Corrected by M. F. Hasler, Dec 05 2013]
A108775(a(n)) = 4.
There are 31092 terms less than 122522399. - T. D. Noe, Dec 04 2013

			27720 is in sequence because sigma(27720) / 27720 = 112320 / 27720 = 4.0519....

Jaroslav Krizek, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A000203, A023198, A023199, A108775.
Cf. A005100 (deficient numbers with abundancy 1 <= a < 2),
Cf. A204829 (numbers with abundancy 2 <= a < 3),
Cf. A204828 (abundant numbers with abundancy 3 <= a < 4).
Cf. A215264 (abundant numbers with abundancy > 5).

Select[Range[200000], 4 <= DivisorSigma[1, #]/# < 5 &] (* T. D. Noe, Dec 04 2013 *)

Corrected and edited by M. F. Hasler, Dec 05 2013

A291459 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 5.

Original entry on oeis.org

294053760, 575134560, 739458720, 882161280, 1193512320, 1314593280, 1725403680, 2539555200, 2588105520, 2646483840, 2711348640, 3008396160, 3891888000, 4053329280, 4214770560, 4648644000, 4802878080, 5176211040, 5194949760, 5258373120, 6470263800, 6768891360, 7900532640

Offset: 1

Paolo P. Lava, Aug 24 2017

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

Subset of A215264.

			One of the proper divisors of 294053760 is 2056320 and sigma(294053760) - 5*2056320 = 1480550400 - 10281600 = 1470268800 = 5*294053760.
One of the proper divisors of 3891888000 is 314496 and sigma(3891888000) - 5*314496 = 19461012480 - 1572480 = 19459440000 = 5*3891888000.

Cf. A000203, A111592, A215264, A291457, A291458.

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(10^10,5);

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

A226589 Numbers k such that sigma(k) >= sigma(k-2) + sigma(k-1) + sigma(k+1) + sigma(k+2).

Original entry on oeis.org

183783600, 232792560, 273873600, 367567200, 410810400, 441080640, 465585120, 547747200, 551350800, 575134560, 612612000, 616215600, 698377680, 735134400, 745945200, 821620800, 845404560, 882161280, 908107200, 931170240, 944863920, 958557600, 1102701600, 1127206080

Offset: 1

Alex Ratushnyak, Jun 12 2013

Among first 1000 terms all are such that sigma(k) > sigma(k-2) + sigma(k-1) + sigma(k+1) + sigma(k+2). - Alex Ratushnyak, Jun 16 2013

			Sigma(183783598...183783602) = {275675400, 194528880, 929940480, 183783602, 275737500}. Because 929940480 > 275675400 + 194528880 + 183783602 + 275737500, 183783600 is in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000203, A215264.

cp's OEIS Frontend

A307115 Primitive 5-abundant numbers: Numbers k such that sigma(k) > 5k (A215264) all of whose proper divisors d are 5-deficient numbers (having sigma(d) < 5d).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A068404 Numbers k such that sigma(k) > 4*k.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A291459 Numbers n having a proper divisor d such that sigma(n) - k*d = k*n. Case k = 5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A226589 Numbers k such that sigma(k) >= sigma(k-2) + sigma(k-1) + sigma(k+1) + sigma(k+2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A291459 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 5.