A087248 -id:A087248 - cp's OEIS frontend

A112643 Odd squarefree abundant numbers.

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245

Offset: 1

Labos Elemer, Sep 20 2005

nonn

Deviates from A046391 (does not contain 36465, 40755 for example).
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 34, 134, 1663, 16328, 175630, 1694621, 16726454, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00016... . - Amiram Eldar, Sep 02 2022
From Amiram Eldar, Jan 15 2025: (Start)
The least term that is not divisible by 5 is a(3696) = 22309287.
The least term that is not divisible by 3 is a(5607800) = 33426748355.
The least term that is coprime to 15 is 1357656019974967471687377449. (End)

			199815 = 3 * 5 * 7 * 11 * 173, with 32 divisors adding up to 400896 = 2 * 199815 + 1266.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A001221, A005101, A005231, A005408, A046391, A087248.

# see A087248 for the additional code
isA112643 := proc(n)
    isA087248(n) and type(n,'odd') ;
end proc:
for n from 1 do
    if isA112643(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

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], Print[n, PrimeFactorList[n], s]; ta = Append[ta, n]], {n, 1, 200000}];{ta = Delete[ta, 1], g}(* Elemer *)
Select[Range[1, 99999, 2], MoebiusMu[#] != 0 && DivisorSigma[1, #] > 2 # &] (* Alonso del Arte, Nov 11 2017 *)

is(n)=if(n%2==0, return(0)); my(f=factor(n)); sigma(f)>2*n && vecmax(f[,2])==1 \\ Charles R Greathouse IV, Feb 21 2017

A087248 INTERSECT A005408.

omega(a(n)) >= 5, where omega(n) = A001221(n) is the number of distinct primes dividing n. - Amiram Eldar, Jan 15 2025

A380929 Numbers k such that A380845(k) > 2*k.

Original entry on oeis.org

36, 72, 84, 140, 144, 168, 180, 264, 270, 280, 288, 300, 336, 360, 372, 392, 450, 520, 528, 532, 540, 558, 560, 576, 594, 600, 612, 620, 672, 720, 744, 756, 780, 784, 840, 900, 930, 1036, 1040, 1050, 1056, 1064, 1068, 1080, 1092, 1116, 1120, 1134, 1152, 1170, 1180, 1188, 1200

Offset: 1

Amiram Eldar, Feb 08 2025

Analogous to abundant numbers (A005101) with A380845 instead of A000203.

			36 is a term since A380845(36) = 84 > 2 * 36 = 72.

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

Cf. A000203, A380845, A380846.
Subsequence of A005101.
Subsequences: A380847, A380848, A380930, A380931.
Similar sequences: A034683, A064597, A087248, A129575, A129656, A292982, A348274, A348604, A379029.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k]; Select[Range[1200], q]

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}

A357695 Cubefree abundant numbers.

Original entry on oeis.org

12, 18, 20, 30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 114, 126, 132, 138, 140, 150, 156, 174, 180, 186, 196, 198, 204, 210, 220, 222, 228, 234, 246, 252, 258, 260, 276, 282, 294, 300, 306, 308, 318, 330, 340, 342, 348, 350, 354, 364, 366, 372, 380, 390, 396

Offset: 1

Amiram Eldar, Oct 10 2022

nonn

The least odd term is a(224) = A357697(1) = 1575.
The lower asymptotic density of this sequence is larger than 12/(91*zeta(3)) = 0.1097... which is the density of its subsequence of cubefree numbers larger than 6 and divisible by 6.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 13, 143, 1440, 14470, 144187, 1442500, 14426015, 144267400, 1442567879, 14425142573, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1442... .

			12 = 2^2 * 3 is a term since it is cubefree and sigma(12) = 28 > 2*12.

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

Intersection of A004709 and A005101.
Subsequences: A087248, A357696, A357697.
Cf. A000203 (sigma), A002117, A308618.

f[p_, e_] := (p^(e+1)-1)/(p-1); q[1] = False; q[n_] := AllTrue[(fct = FactorInteger[n])[[;;, 2]], # < 3 &] && Times @@ f @@@ fct > 2*n; Select[Range[400], q]

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

A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.

Original entry on oeis.org

1, 2, 5, 13, 26, 29, 37, 41, 61, 73, 74, 101, 109, 113, 122, 137, 146, 157, 173, 181, 193, 218, 229, 257, 277, 281, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 541, 554, 569, 601, 613, 617, 626, 641, 653, 661, 673, 677

Offset: 1

Labos Elemer, Oct 29 2001

nonn

From Amiram Eldar, Mar 08 2025: (Start)
Number k such that A280710(k) * A280710(A000203(k)) = 1, or equivalently, A280710(k) * A280710(A048250(k)) = 1.
Squarefree numbers k whose prime factors are terms of A049097, and the elements of the set {p+1 , p|k} are pairwise coprime. (End)

			For k = 13, sigma(13) = 14 = 2*7 is squarefree.
For k = 26, sigma(26) = 1 + 2 + 13 + 26 = 42 = 2*3*7 is squarefree.
For k = 277 (prime), sigma(277) = 278 = 2*139 is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A008683, A048250, A065299, A065300.

Cf. A049097, A065303, A087248, A087249, A280710.

Select[Range[1000],AllTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* Harvey P. Dale, Aug 09 2014 *)

is(m) = abs(moebius(m))==1 && abs(moebius(sigma(m)))==1 \\ Harry J. Smith, Oct 15 2009

from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n)==n
print([n for n in range(1, 1001) if issquarefree(n) and issquarefree(divisor_sigma(n,1))]) # Indranil Ghosh, Mar 19 2017

A087247 Squarefree deficient nonprime numbers.

Original entry on oeis.org

1, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 105, 106, 110, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161, 165, 166, 170, 177, 178, 182, 183

Offset: 1

Labos Elemer, Sep 05 2003

nonn

			105 = 3*5*7 and sigma(105) = 1 + 3 + 5 + 7 + 15 + 21 + 35 + 105 = 192 < 210 = 2*105, so 105 is in the sequence.
The sequence differs from A006881: first term with 3 distinct prime factors is 105.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Cf. A087244, A087245, A087246, A087248.

Cf. A008683, A000203, A013929, A005117, A005100, A005101.

isA087247[n_] := SquareFreeQ[n] && !PrimeQ[n] && (DivisorSigma[1, n] < 2n); Select[Range[200], isA087247] (* Enrique Pérez Herrero, Jan 13 2011 *)

isok(n) = !isprime(n) && issquarefree(n) && (sigma(n) < 2*n); \\ Michel Marcus, Jul 09 2018

A249242 Squarefree primitive abundant numbers (using the second definition: having no abundant proper divisors, cf. A091191).

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398, 1430, 1434, 1446, 1506

Offset: 1

Derek Orr, Oct 23 2014

nonn

Primitive numbers in A087248.
Squarefree numbers in A091191.
According to the definition of A091191, all terms of the form 6*p, p > 3, are in this sequence (and similarly for other perfect numbers). Primitive abundant can also be defined as "having only deficient proper divisors", cf. A071395. The corresponding squarefree terms are listed in A298973, and those with n prime factors are counted in A295369. (The preceding remark shows that this count would be infinite for n = 3, using the definition of A091191.) - M. F. Hasler, Feb 16 2018

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

Intersection of A087248 and A091191.

Cf. A071395, A295369, A298973.

Select[Range@1506, SquareFreeQ[#] && DivisorSigma[1, #] > 2 #  && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &] (* Amiram Eldar, Jun 26 2019 after Michael De Vlieger at A091191 *)

v=[];for(n=1,10^5,d=0;for(j=2,ceil(sqrt(n)),if(n%(j^2),d++));if(d==ceil(sqrt(n))-1,if(sigma(n)>2*n,c=0;for(i=1,#v,if(n%v[i],c++));if(c==#v,print1(n,", ");v=concat(v,n)))))

Definition edited by M. F. Hasler, Feb 16 2018

A357686 Nonsquarefree numbers k such that A293228(k) > k.

Original entry on oeis.org

60, 84, 132, 140, 156, 204, 228, 276, 348, 372, 420, 444, 492, 516, 564, 636, 660, 708, 732, 780, 804, 852, 876, 924, 948, 996, 1020, 1068, 1092, 1140, 1164, 1212, 1236, 1284, 1308, 1356, 1380, 1428, 1524, 1540, 1572, 1596, 1644, 1668, 1716, 1740, 1788, 1812, 1820

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

The squarefree numbers k such that A293228(k) > k are the squarefree abundant numbers (A087248).
If k > 3 is a term of A243128 then 4*k is a term.
The least odd term is (3/2)*prime(17)# = 2884140525231318958605.
The least term that is coprime to 6 is (5/6)*prime(1245)# = 5.629...*10^4361.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 2, 26, 287, 2725, 27660, 275298, 2754638, 27556849, 275538900, 2755151247, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02755... .

			60 = 2^2 * 15 is a term since it is nonsquarefree, its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

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

Intersection of A013929 and A357685.
Subsequence of A005101.
Cf. A087248, A243128.

q[n_] := AnyTrue[(f = FactorInteger[n])[[;;, 2]], # > 1 &] && Times @@ (1 + f[[;; , 1]]) > n; Select[Range[2, 2000], q]

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

A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 120, 138, 150, 168, 174, 186, 210, 222, 246, 258, 270, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822

Offset: 1

Amiram Eldar, Dec 14 2024

All the squarefree abundant numbers (A087248) are terms since A241405(k) = A000203(k) for a squarefree number k.
If k is a term and m is coprime to k them k*m is also a term.
The numbers of terms that do no exceed 10^k, for k = 2, 3, ..., are 5, 67, 767, 7595, 76581, 764321, 7644328, 76468851, 764630276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07646... .

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

Cf. A000203, A241405, A323757, A323758, A323759, A379027.
Subsequence of A005101.
Subsequences: A034683, A087248, A379030, A379031.
Similar sequences: A064597, A129575, A129656, A292982, A348274, A348604.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1000], meAbQ]

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

A087249 Squarefree numbers k such that sigma(k) is not squarefree.

Original entry on oeis.org

3, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 39, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 62, 65, 66, 67, 69, 70, 71, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 102, 103, 105, 106, 107, 110, 111, 114, 115, 118, 119, 123, 127, 129, 130

Offset: 1

Labos Elemer, Sep 05 2003

nonn

			For k=7: sigma(7) = 2*2*2 = 8.
For k=10: sigma(10) = 1 + 2 + 5 + 10 = 18 = 2*3*3.

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

Complement of A065301 within A005117.

Cf. A065303, A087248.

Select[Range[150],SquareFreeQ[#]&&!SquareFreeQ[DivisorSigma[1,#]]&]  (* Harvey P. Dale, Feb 06 2011 *)

is(k) = issquarefree(k) && !issquarefree(sigma(k)); \\ Amiram Eldar, Jun 15 2024

A357685 Numbers k such that A293228(k) > k.

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 102, 114, 132, 138, 140, 156, 174, 186, 204, 210, 222, 228, 246, 258, 276, 282, 318, 330, 348, 354, 366, 372, 390, 402, 420, 426, 438, 444, 462, 474, 492, 498, 510, 516, 534, 546, 564, 570, 582, 606, 618, 636, 642, 654, 660, 678, 690

Offset: 1

Amiram Eldar, Oct 09 2022

nonn

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 7, 79, 843, 8230, 83005, 826875, 8275895, 82790525, 827718858, 8276571394, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0827... .

			30 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15} and their sum is 42 > 30.
60 is a term since its aliquot squarefree divisors are {1, 2, 3, 5, 6, 10, 15, 30} and their sum is 72 > 60.

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

Cf. A005117, A293228.
Disjoint union of A087248 and A357686.
Subsequence of A005101.
Similar sequences: A034683, A064597, A129575, A129656, A292982, A348274, A348604.

s[n_] := Times @@ (1 + (f = FactorInteger[n])[[;; , 1]]) - If[AllTrue[f[[;;, 2]], # == 1 &], n, 0]; Select[Range[2, 1000], s[#] > # &]

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

cp's OEIS Frontend

A112643 Odd squarefree abundant numbers.

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A380929 Numbers k such that A380845(k) > 2*k.

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A357695 Cubefree abundant numbers.

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A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.

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A087247 Squarefree deficient nonprime numbers.

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A249242 Squarefree primitive abundant numbers (using the second definition: having no abundant proper divisors, cf. A091191).

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A357686 Nonsquarefree numbers k such that A293228(k) > k.

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