A087248 -id:A087248 - cp's OEIS frontend

A381740 Squarefree numbers k such that k^2 is abundant.

6, 10, 14, 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 170, 174, 182, 186, 190, 210, 222, 230, 238, 246, 258, 266, 282, 286, 290, 310, 318, 322, 330, 354, 366, 370, 374, 390, 402, 406, 410, 418, 426, 430, 434, 438, 442, 462, 470, 474, 498, 506, 510

Offset: 1

Amiram Eldar, Mar 05 2025

The least odd term is a(105) = 105.
The least term that is coprime to 6 is a(3637276) = 37182145.
If k is a squarefree number that is divisible by 6, 10 or 14, then it is a term. Therefore a lower bound for the asymptotic density of this sequence is 29/(192*zeta(2)) = 0.0918... .
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 2, 8, 99, 972, 9826, 97610, 979190, 9770801, 97650638, 976893969, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0976... .
If k is a term then any multiple of k that is squarefree is a term. The primitive terms are in A381741.

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

Intersection of A005117 and A381738.
Subsequences: A087248, A381741.
Cf. A013661.

Select[Range[600], SquareFreeQ[#] && DivisorSigma[-1, #^2] > 2 &]

isok(k) = {my(f = factor(k)); if(!issquarefree(f), 0, prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2);}

A112644 Odd and squarefree abundant numbers not divisible by 5.

Original entry on oeis.org

22309287, 28129101, 30069039, 34051017, 35888853, 36399363, 38057019, 39768729, 40681641, 41708667, 43444401, 45588543, 45894849, 48141093, 48555507, 50489439, 51294243, 51408357, 53804751, 54777723, 55186131, 56429373, 57228171, 58555497, 59168109

Offset: 1

Labos Elemer, Sep 20 2005

nonn

The least term that is not divisible by 3 is 73#/5# = Product_{k=4..21} prime(k) = 1357656019974967471687377449. - Amiram Eldar, Aug 15 2024

			99906807 = 3*7*11*13*17*19*103 is a term since it is an odd squarefree number that is not divisible by 5, and sigma(99906807) = 201277440 > 2*99906807.

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

Cf. A087248, A046391, A112643, A112642, A112640.

Cf. A005231, A047802.

ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&Equal[Abs[MoebiusMu[n]], 1]&& !Equal[Mod[n, 2], 0]&&!Equal[Mod[n, 5], 0], Print[n, PrimeFactorList[n], s];ta=Append[ta, n]], {n, 10000000, 100000000}];{ta=Delete[ta, 1], g}

issfab(k) = my(f = factor(k)); issquarefree(f) && sigma(f, -1) > 2;
is(k) = gcd(k, 10) == 1 && issfab(k); \\ Amiram Eldar, Aug 15 2024

A379030 Nonsquarefree modified exponential abundant numbers: nonsquarefree numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

120, 150, 168, 270, 294, 420, 630, 660, 726, 750, 780, 840, 924, 990, 1014, 1020, 1050, 1092, 1140, 1170, 1320, 1380, 1386, 1428, 1470, 1530, 1560, 1596, 1638, 1650, 1710, 1734, 1740, 1848, 1860, 1890, 1950, 2040, 2058, 2070, 2142, 2166, 2184, 2220, 2280, 2394

Offset: 1

Amiram Eldar, Dec 14 2024

All the squarefree abundant numbers (A087248) are also modified exponential abundant numbers (A379029). This sequence lists the terms of A379029 that are not in A087248.

The numbers of terms that do no exceed 10^k, for k = 3, 4, ..., are 14, 211, 2090, 21236, 212744, 2123071, 21235175, 212450318, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0212... .

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

Intersection of A013929 and A379029.
Complement of A087248 within A379029.
Cf. A005117, A241405.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; nsmeAbQ[n_] := !SquareFreeQ[n] && mesigma[n] > 2*n; Select[Range[2400], nsmeAbQ]

is(n) = {my(f=factor(n)); if(issquarefree(f), 0, prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*n);}

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.

A354951 The number of squarefree abundant numbers whose largest prime divisor is prime(n).

Original entry on oeis.org

0, 0, 1, 3, 5, 13, 27, 59, 125, 257, 534, 1094, 2244, 4607, 9407, 19164, 38939, 79154, 160473, 325213, 658940, 1332598, 2694009, 5440236, 10973874, 22135366, 44644593, 89989907, 181374472, 365371286, 734959945

Offset: 1

Amiram Eldar, Jun 13 2022

For n >= 3 we have a(n) >= 2^(n-3) as all squarefree numbers whose largest prime divisor are prime(n) that are a multiple of 6*prime(n) are abundant. - David A. Corneth, Jun 13 2022

			n  prime(n)  a(n)  terms k of A087248 with A006530(k) = prime(n)
-  --------  ----  ---------------------------------------------
1  2         0     -
2  3         0     -
3  5         1     30
4  7         3     42, 70, 210
5  11        5     66, 330, 462, 770, 2310

Cf. A005117, A006530, A087248, A354950.

a[n_] := Count[Prime[n] * Divisors[Product[Prime[i], {i, 1, n - 1}]], _?(SquareFreeQ[#] && DivisorSigma[-1, #] > 2 &)]; Array[a, 15]

A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 79170, 82110, 84630, 85470, 91770, 94710, 99330, 101010, 103530, 108570, 111930, 117390, 122430, 128310, 136290, 140910, 144690, 154770, 161070, 164010, 166530, 168630, 182490, 191730, 205590

Offset: 1

Amiram Eldar, Aug 19 2025

nonn

First differs from A067885 at n = 11: A067885(11) = 72930 is not a term of this sequence. a(59) = 510510 is the least term of this sequence that is not in A067885.
Subsequence of A285615 and first differs from it at n = 51: A285615(51) = 390390 is not a term of this sequence.
This sequence is not the same as the sequence of numbers k such that A048250(k) > 3*k which includes all the terms of this sequence but also nonsquarefree numbers, the least of them is 2*A002110(52) = A088860(52) = 2.1248...*10^96.
The least odd term is A002110(17)/2 = 961380175077106319535, the least term that is not divisible by 3 is a(5607800) = 66853496710, and the least term that is coprime to 6 is A002110(52)/6 = 1.7706...*10^95.
If k is a term and m is a squarefree number coprime to k, then k*m is also a term.
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 17, 95, 795, 8162, 86331, 854164, 8372782, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00008... .

			30030 = 2 * 3 * 5 * 7 * 11 * 13 is a term since it is squarefree, and sigma(30030) = 96768 > 3*30030 = 90090.

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

Intersection of A005117 and A068403.
Subsequence of A087248 and A285615.
Cf. A000203, A001221, A002110, A013929, A048250, A088860.

q[k_] := Module[{f = FactorInteger[k]}, Max[f[[;;, 2]]] == 1 && Times @@ (1 + f[[;; , 1]]) > 3*k]; Select[Range[2*10^5], q]

isok(k) = {my(f = factor(k)); issquarefree(f) && vecprod(apply(x -> x+1, f[, 1])) > 3*k;}

A001221(a(n)) >= 6.

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.

A357687 Nonsquarefree numbers k such that A048250(k) > 2*k.

Original entry on oeis.org

401120980260, 14841476269620, 16445960190660, 17248202151180, 18852686072220, 608500527054420, 638183479593660, 697549384672140, 707176288198380, 772960128961020, 810665501105460, 26165522663340060, 28599524771557740, 29994623540902020, 33237285545323860, 1229779565176982820

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

The squarefree numbers k such that A048250(k) > 2*k are the squarefree abundant numbers (A087248).

The least odd term is 3*prime(553)#/2 = 3.735...*10^1709.

Subsequence of A005101 and A013929.

Cf. A005117, A048250, A087248, A357685, A357686.

q[n_] := Module[{f = FactorInteger[n]}, AnyTrue[f[[;;, 2]], # > 1 &] && Times @@ (f[[;;, 1]]+1) > 2*n];

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

cp's OEIS Frontend

A381740 Squarefree numbers k such that k^2 is abundant.

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A112644 Odd and squarefree abundant numbers not divisible by 5.

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A379030 Nonsquarefree modified exponential abundant numbers: nonsquarefree numbers k such that A241405(k) > 2*k.

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

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A354951 The number of squarefree abundant numbers whose largest prime divisor is prime(n).

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A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

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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.

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A357687 Nonsquarefree numbers k such that A048250(k) > 2*k.

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