A357695 -id:A357695 - 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};

A357696 Cubefree primitive abundant numbers: cubefree abundant numbers having no abundant proper divisor.

Original entry on oeis.org

12, 18, 20, 30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 196, 222, 246, 258, 282, 308, 318, 354, 364, 366, 402, 426, 438, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822, 834, 836, 868, 894, 906, 942, 978, 1002

Offset: 1

Amiram Eldar, Oct 10 2022

nonn

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

Intersection of A004709 and A091191.
Subsequence of A357695.
A249242 is a subsequence.
Cf. A308618.

cubeFreeQ[n_] := Max[FactorInteger[n][[;; , 2]]] < 3; primQ[n_] := DivisorSigma[-1, n] > 2 && AllTrue[n/FactorInteger[n][[;; , 1]], DivisorSigma[-1, #] <= 2 &]; Select[Range[1500], cubeFreeQ[#] && primQ[#] &]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] > 2, return(0))); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, if(sigma(n/f[i,1], -1) > 2, return(0))); 1};

A357697 Odd cubefree abundant numbers.

Original entry on oeis.org

1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11025, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 17325, 18585, 19215, 19635, 20475, 21105, 21945, 22365, 22995, 23205, 24255, 24885, 25935, 26145, 26565, 26775

Offset: 1

Amiram Eldar, Oct 10 2022

nonn

First differs from A333950 at n = 1258. Terms that are not in A333950 include 8564325, 8565795, 8567325, ... and terms of A333950 that are not here include 1126125, 2096325, 2207205, ... .

The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 16, 125, 1127, 11734, 116911, 1162781, 11638566, 116342286, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00116... .

			1575 = 3^2 * 5^2 * 7 is a term since it is odd and cubefree and sigma(1575) = 3224 > 2*1575.

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

Intersection of A004709 and A005231.
Intersection of A005408 and A357695.
A112643 is a subsequence.
Cf. A000203 (sigma), A333950.

f[p_, e_] := (p^(e+1)-1)/(p-1); q[1] = 0; q[n_] := AllTrue[(fct = FactorInteger[n])[[;;, 2]], # < 3 &] && Times @@ f @@@ fct > 2*n; Select[Range[1, 30000, 2], q]

is(n) = {my(f); if(n%2 == 0, return(0)); f = factor(n); (n==1 || vecmax(f[,2]) < 3) && sigma(f, -1) > 2};

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.

A357699 Noncubefree numbers k such that A357698(k) > k.

Original entry on oeis.org

24, 40, 72, 120, 168, 200, 264, 280, 312, 360, 392, 408, 440, 456, 504, 520, 540, 552, 600, 616, 680, 696, 728, 744, 760, 792, 840, 888, 920, 936, 952, 984, 1032, 1064, 1128, 1144, 1160, 1176, 1224, 1240, 1272, 1288, 1320, 1368, 1400, 1416, 1464, 1480, 1496, 1560

Offset: 1

Amiram Eldar, Oct 10 2022

nonn

The cubefree numbers k such that A357698(k) > k are the cubefree abundant numbers (A357695).
The least odd term is (3/4) * prime(4)# * prime(11)# = 31588277195475.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 3, 32, 319, 3256, 32404, 323837, 3243328, 32425481, 324212022, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0324... .

			24 = 2^3 * 3 is a term since it is divisible by a cube and A357698(24) = 28 > 24.

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

Cf. A046099, A357695, A357698.

f[p_, e_] := 1 + p + If[e == 1, 0, p^2]; q[n_] := AnyTrue[(fct = FactorInteger[n])[[;;, 2]], # > 2 &] && Times @@ f @@@ fct > n; Select[Range[2, 2000], 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)) > n};

A387155 The number of n-free abundant numbers below 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

22148167706, 52012, 10828, 24601, 23660, 29114, 58967, 118828, 238600, 478099, 957324, 1916191, 3834167, 7669094, 15335488, 30667762, 61337894, 122679755, 245357929, 490718137, 981456651, 1962956352, 3925957422, 7851819466, 15703524589, 31406984903, 62813576969

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, respectively.

			a(2) = 22148167706 because there are 22148167706 squarefree numbers k such that A048250(k) > 2*k (i.e., terms of A087248) that are less than the least nonsquarefree number k that has this property, A387154(2) = 401120980260.
a(3) = 52012 because there are 52012 cubefree numbers k such that A073185(k) > 2*k (i.e., terms of A357695) that are less than the least noncubefree number k that has this property, A387154(3) = 360360.

Cf. A004709, A005117, A046099, A048250, A073185, A091194, A302991, A357695, A357700, A387154.

freeQ[n_, k_] := AllTrue[FactorInteger[n][[;; , 2]], # < k &];
sigma[n_, k_] := Times @@ ((First[#]^(Min[Last[#], k - 1] + 1) - 1)/(First[#] - 1) & /@ FactorInteger[n]);
a[n_] := Module[{m = 2, c = 0}, While[True, If[sigma[m, n] > 2*m, c++; If[!freeQ[m, n], Break[]]]; m++]; c-1];

isfree(n, k) = if(n == 1, 1, my(e = factor(n)[,2]); for(i=1, #e, if(e[i] >= k, return(0))); 1);
sigmafree(n, k) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(min(f[i,2],k-1)+1)-1)/(f[i,1]-1));}
a(n) = {my(m = 2, c = 0); while(1, if(sigmafree(m, n) > 2*m, c++; if(!isfree(m, n), break)); m++); c-1;}

Let A_k(n) be the number of k-free abundant numbers that are not exceeding n. Then, a(n) = A_n(A387154(n)) - 1.

a(n) ~ c * 945 * 2^n, where c = A302991.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A357696 Cubefree primitive abundant numbers: cubefree abundant numbers having no abundant proper divisor.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A357697 Odd cubefree abundant numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A357699 Noncubefree numbers k such that A357698(k) > k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A387155 The number of n-free abundant numbers below the least number k that is not n-free whose sum of n-free divisors is larger than 2*k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula