A073185 -id:A073185 - cp's OEIS frontend

A357700 Noncubefree numbers k such that A073185(k) > 2*k.

360360, 471240, 1801800, 2356200, 2522520, 2633400, 2784600, 3112200, 3187800, 3298680, 3686760, 3767400, 3898440, 3963960, 4019400, 4296600, 4462920, 4684680, 5128200, 5183640, 5682600, 5793480, 6126120, 6846840, 8011080, 8288280, 8953560, 10210200, 10450440

Offset: 1

Amiram Eldar, Oct 10 2022

nonn

The cubefree numbers k such that A073185(k) > 2*k are the cubefree abundant numbers (A357695).

The least odd term is (3/4) * prime(13)# * prime(197)# = 6.252...*10^517.

			360360 = 2^3 * 45045 is a term since it is divisible by a cube and A073185(360360) = 733824 > 2*360360.

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

Cf. A073185, A357695.

f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; q[n_] := AnyTrue[(fct = FactorInteger[n])[[;;, 2]], # > 2 &] && Times @@ f @@@ fct > 2*n; Select[Range[2, 5*10^6], q]

is(n) = {my(f = factor(n)); if(n == 1 || vecmax(f[,2]) < 3, return(0)); prod(i=1, #f~, 1 + f[i,1] + if(f[i,2]==1, 0, f[i,1]^2)) > 2*n};

A073184 Number of cubefree divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 3, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 6, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 6, 3, 4, 2, 12, 4, 4, 4, 6, 2, 12, 4, 6, 4, 4, 4, 6, 2, 6, 6, 9, 2, 8, 2

Offset: 1

Reinhard Zumkeller, Jul 19 2002

a(n) = number of divisors of the cubefree kernel of n: a(n) = A000005(A007948(n)); [corrected by Amiram Eldar, Oct 08 2022]

Multiplicative because it is the Inverse Möbius transform of the characteristic function of cubefree numbers. a(n) is a prime signature sequence. a(p) = 2, a(p^e) = 3, e>1. - Christian G. Bower, May 18 2005

			The divisors of 56 are {1, 2, 4, 7, 8, 14, 28, 56}, 8=2^3 and 56=7*2^3 are not cubefree, therefore a(56) = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000012, A001620, A004709, A027750, A034444, A073180, A073183, A073185, A212793.

a073184 = sum . map a212793 . a027750_row
-- Reinhard Zumkeller, May 27 2012

a[1] = 1; a[p_?PrimeQ] = 2; a[n_] := Times @@ (If[#[[2]] == 1, 2, 3] & /@ FactorInteger[n]); Table[a[n], {n, 1, 103}] (* Jean-François Alcover, May 24 2012, after Christian G. Bower *)

a(n) = {my(e = factor(n)[,2]); prod(i = 1, #e, if(e[i] == 1, 2, 3))}; \\ Amiram Eldar, Oct 08 2022

a(n) <= A073182(n).
Dirichlet g.f.: zeta(s)^2/zeta(3*s). Dirichlet convolution of the characteristic function of cubefree numbers by A000012. - R. J. Mathar, Apr 12 2011
a(n) = Sum_{k = 1..A000005(n)} A212793(A027750(n,k)). - Reinhard Zumkeller, May 27 2012
Sum_{k=1..n} a(k) ~ n / zeta(3) * (log(n) - 1 + 2*gamma - 3*zeta'(3)/zeta(3)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019

A385006 The sum of the biquadratefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 15, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 15, 84, 144

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A365682 and A366992 at n = 32.

The number of these divisors is A252505(n), and the largest of them is A058035(n).

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

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

A073183 Sum of divisors of n that are not greater than the cubefree kernel of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 36, 31, 42, 13, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 50, 42, 96, 44, 84, 78, 72, 48, 36, 57, 93, 72, 98, 54, 39, 72, 64, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68, 126, 96, 144, 72

Offset: 1

Reinhard Zumkeller, Jul 19 2002

a(n) >= A073185(n).

			The cubefree kernel of 56 = 7 * 2^3 is 28 = 7 * 2^2 and the divisors <= 28 of 56 are {1, 2, 4, 7, 8, 14, 28}, therefore a(56) = 1 + 2 + 4 + 7 + 8 + 14 + 28 = 64.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A007948, A073182, A073181.

sdcfk[n_]:=Module[{cf=Times@@Flatten[Table[#[[1]],#[[2]]]&/@({#[[1]],If[ #[[2]]>2,2,#[[2]]]}&/@FactorInteger[n])]},Total[Select[Divisors[n],#<= cf&]]]; Array[sdcfk,80] (* Harvey P. Dale, Jul 14 2018 *)

a007948(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = min(f[i, 2], 2)); factorback(f);
a(n) = sumdiv(n, d, d*(d<=a007948(n))); \\ Michel Marcus, Feb 07 2015

A308618 Cubefree superabundant numbers: cubefree numbers (A004709) k such that sigma(k)/k > sigma(j)/j for all cubefree numbers j < k.

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 420, 1260, 4620, 6300, 13860, 69300, 180180, 900900, 3063060, 15315300, 58198140, 290990700, 1338557220, 2036934900, 6692786100, 38818159380, 46849502700, 194090796900, 1358635578300, 6016814703900, 42117702927300, 222622144044300

Offset: 1

Amiram Eldar, Aug 21 2019

nonn

Erdős and Nicolas named these numbers "nombres sans cube superabondants".
All the terms are either primorials (A002110) or products of two primorials.
Also numbers m such that A073185(m)/m > A073185(k)/k for all k < m. - Amiram Eldar, Oct 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..1307
Paul Erdős and Jean-Louis Nicolas, Répartition des nombres superabondants", Bulletin de la Société Mathématique de France, Vol. 103 (1975), pp. 65-90. See section 5, p. 83.

Cf. A000203, A002110, A004394, A004709, A073185.

Subsequence of A025487 and A220423.

cubeFreeQ[n_] := Max @ FactorInteger[n][[;;, 2]] < 3; s = {}; rm = 0; Do[If[ !cubeFreeQ[n], Continue[]]; r = DivisorSigma[1, n]/n; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

A385005 The sum of the cubefull divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 57, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 121, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 25, 109, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 15 2025

The sum of the terms in A036966 that divide n.

The number of these divisors is A190867(n), and the largest of them is A360540(n).

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

Cf. A036966, A190867, A183097, A360540.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), this sequence (cubefull), A385006 (biquadratefree).

f[p_, e_] := (p^(e+1)-1)/(p-1) - p - If[e == 1, 0, p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^(e+1)-1)/(p-1) - p - if(e == 1, 0, p^2));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = ((p^(e+1)-1) / (p-1)) - p - p^2 if e >= 3.

Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - p^(s-1) + 1/p^(3*s-3)).

A384554 The sum of the infinitary divisors of n that are cubefree.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 7, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 28, 26, 42, 13, 40, 30, 72, 32, 3, 48, 54, 48, 50, 38, 60, 56, 42, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 39, 72, 56, 80, 90, 60, 120, 62, 96, 80, 5, 84, 144, 68, 90

Offset: 1

Amiram Eldar, Jun 03 2025

The number of these divisors is A368883(n), and the largest of them is A384555(n).

The sum of the infinitary divisors of n that are squarefree (A005117) is A367991(n).

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

Cf. A000203, A005117, A004709, A013666, A049417, A073185, A077609, A367991, A368883, A384555.

f[p_, e_] := Switch[Mod[e, 4], 0, 1, 1, p+1, 2, p^2+1, 3, p^2+p+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; [1, p+1, p^2+1, p^2+p+1][e%4+1]);}

from math import prod
from sympy import factorint
def A384554(n): return prod((1,p+1,p**2+1,p*(p+1)+1)[e&3] for p,e in factorint(n).items()) # Chai Wah Wu, Jun 03 2025

Multiplicative with a(p^e) = 1 if e == 0 (mod 4), p + 1 if e == 1 (mod 4), p^2 + 1 if e == 2 (mod 4), and p^2 + p + 1 if e == 3 (mod 4).
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
a(n) <= A049417(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s + 1/p^(2*s-2) + 1/p^(2*s) + 1/p^(3*s-1) + 1/p^(3*s-2) + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(8) * Product_{p prime} (1 + 1/p^2 - 2/p^3 + 2/p^4 - 1/p^5 - 1/p^7) = 1.2351002232125595782019... .

A357698 a(n) is the sum of the aliquot divisors of n that are cubefree.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 7, 1, 21, 1, 22, 11, 14, 1, 28, 6, 16, 13, 28, 1, 42, 1, 7, 15, 20, 13, 55, 1, 22, 17, 42, 1, 54, 1, 40, 33, 26, 1, 28, 8, 43, 21, 46, 1, 39, 17, 56, 23, 32, 1, 108, 1, 34, 41, 7, 19, 78, 1, 58, 27, 74, 1, 91, 1, 40

Offset: 1

Amiram Eldar, Oct 10 2022

nonn

			The divisors of 16 that are cubefree are {1, 2, 4}, and their sum is a(16) = 1 + 2 + 4 = 7.

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

Cf. A004709, A073185, A212793, A293228.

Cf. A013661, A002117.

f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; a[1] = 0; a[n_] := Times @@ f @@@ (fct = FactorInteger[n]) - If[AllTrue[fct[[;;, 2]], # < 3 &], n, 0]; Array[a, 100]

a(n) = {my(f = factor(n), s); s = prod(i=1, #f~, 1 + f[i,1] + if(f[i,2] == 1, 0, f[i,1]^2)); if(n==1 || vecmax(f[,2]) < 3, s -= n); s};

a(n) = Sum_{d|n, dA212793(d)*d.
a(n) = A073185(n) - (A212793(n)*n).
a(n) = 1 iff n is a prime.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2) - 1)/(2*zeta(3)) = 0.268262... .

A387154 The least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Original entry on oeis.org

401120980260, 360360, 55440, 110880, 100800, 120960, 241920, 483840, 967680, 1935360, 3870720, 7741440, 15482880, 30965760, 61931520, 123863040, 247726080, 495452160, 990904320, 1981808640, 3963617280, 7927234560, 15854469120, 31708938240, 63417876480, 126835752960

Offset: 2

Amiram Eldar, Aug 19 2025

nonn

n-free numbers are numbers that are not divisible by an n-th power larger than 1. E.g., A005117, A004709, and A046100 for n = 2, 3, and 4, respectively.
The sum of n-free divisors of a number is the sum of its divisors that are n-free numbers. E.g., A048250, A073185, and A385006 for n = 2, 3, and 4, respectively.
All the terms are in A025487.

			For n = 2, the numbers k such that A048250(k) > 2*k include all the squarefree abundant numbers (A087248). The least nonsquarefree number (A013929) k such that A048250(k) > 2*k is 401120980260 = 2^2*3*5*7*11*13*17*19*23*29*31.
For n = 3, the numbers k such that A073185(k) > 2*k include all the cubefree abundant numbers (A357695). The least noncubefree number (A046099) k such that A073185(k) > 2*k is A357700(1) = 360360 = 2^3*3^2*5*7*11*13.

Amiram Eldar, Table of n, a(n) for n = 2..1000

Cf. A004709, A005117, A013929, A025487, A046099, A046100, A048250, A087248, A073185, A357695, A357700, A385006, A387155.

a[n_] := If[n < 7, {401120980260, 360360, 55440, 110880, 100800}[[n-1]], 945 * 2^n]; Array[a, 26, 2]

a(n) = if(n < 7, [401120980260, 360360, 55440, 110880, 100800][n-1], 945 * 2^n);

a(n) = 945 * 2^n for n >= 7.

A133477 Sum of cubefree divisors of n excluding 1.

Original entry on oeis.org

0, 2, 3, 6, 5, 11, 7, 6, 12, 17, 11, 27, 13, 23, 23, 6, 17, 38, 19, 41, 31, 35, 23, 27, 30, 41, 12, 55, 29, 71, 31, 6, 47, 53, 47, 90, 37, 59, 55, 41, 41, 95, 43, 83, 77, 71, 47, 27, 56, 92, 71, 97, 53, 38, 71, 55, 79, 89, 59, 167, 61, 95, 103, 6, 83, 143, 67, 125, 95, 143, 71, 90

Offset: 1

Jonathan Vos Post, Nov 29 2007

			a(8) = 6 because the divisors of 8 are {1,2,4,8}, the cubefree divisors are 1, 2, 4 so we get a(8) = 2 + 4 = 6.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A004709, A005117, A039653, A048250, A068318, A083681.

Cf. A073184, A073185.

scfd[n_]:=Total[Select[Divisors[n],Max[Transpose[FactorInteger[#]][[2]]]<3&]]; Array[scfd,80]-1 (* Harvey P. Dale, Nov 30 2014 *)
f[p_, e_] := 1 + p + If[e > 1, p^2, 0]; a[1] = 0; a[n_] := -1 + Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

a(n) = A073185(n) - 1. - N. J. A. Sloane, Nov 30 2007

Edited by N. J. A. Sloane, Nov 30 2007

cp's OEIS Frontend

A357700 Noncubefree numbers k such that A073185(k) > 2*k.

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A073184 Number of cubefree divisors of n.

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A385006 The sum of the biquadratefree divisors of n.

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A073183 Sum of divisors of n that are not greater than the cubefree kernel of n.

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A308618 Cubefree superabundant numbers: cubefree numbers (A004709) k such that sigma(k)/k > sigma(j)/j for all cubefree numbers j < k.

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A385005 The sum of the cubefull divisors of n.

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A384554 The sum of the infinitary divisors of n that are cubefree.

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A357698 a(n) is the sum of the aliquot divisors of n that are cubefree.