A087247 -id:A087247 - cp's OEIS frontend

A087248 Squarefree abundant numbers.

30, 42, 66, 70, 78, 102, 114, 138, 174, 186, 210, 222, 246, 258, 282, 318, 330, 354, 366, 390, 402, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 642, 654, 678, 690, 714, 762, 770, 786, 798, 822, 834, 858, 870, 894, 906, 910, 930, 942, 966, 978

Offset: 1

Labos Elemer, Sep 05 2003

nonn

First odd term is 15015 = 3 * 5 * 7 * 11 * 13, with 32 divisors that add up to 32256 = 2*15015 + 2226. See A112643. - Alonso del Arte, Nov 06 2017
The lower asymptotic density of this sequence is larger than 1/(2*Pi^2) = 0.05066... which is the density of its subsequence of squarefree numbers larger than 6 and divisible by 6. The number of terms below 10^k for k=1,2,... is 0, 5, 53, 556, 5505, 55345, 551577, 5521257, 55233676, 552179958, 5521420147, ..., so it seems that this sequence has an asymptotic density which equals to about 0.05521... - Amiram Eldar, Feb 13 2021
The asymptotic density of this sequence is larger than 0.0544 (Wall, 1970). - Amiram Eldar, Apr 18 2024

			Checking that 30 = 2 * 3 * 5 and sigma(30) = 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72, which is more than twice 30, we verify that 30 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Charles Robert Wall, Topics related to the sum of unitary divisors of an integer, Ph.D. diss., University of Tennessee, 1970.

Cf. A087244, A087245, A087246, A087247, A087248, A008683, A005117, A005101, A112643.

isA005101 := proc(n)
    simplify(numtheory[sigma](n)>2*n);
end proc:
isA087248 := proc(n)
    isA005101(n) and numtheory[issqrfree](n) ;
end proc:
for n from 1 to 500 do
    if isA087248(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

Select[Range[10^3], SquareFreeQ@ # && DivisorSigma[1, #] > 2 # &] (* Michael De Vlieger, Feb 05 2017 *)

isA087248(i) = (sigma(i) > 2*i) && issquarefree(i) \\ Michel Marcus, Mar 09 2013

A005117 INTERSECT A005101.

A087244 Nonsquarefree deficient numbers.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 44, 45, 49, 50, 52, 63, 64, 68, 75, 76, 81, 92, 98, 99, 116, 117, 121, 124, 125, 128, 135, 136, 147, 148, 152, 153, 164, 169, 171, 172, 175, 184, 188, 189, 207, 212, 225, 232, 236, 242, 243, 244, 245, 248, 250, 256, 261, 268, 275, 279

Offset: 1

Labos Elemer, Sep 05 2003

nonn

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 3, 21, 197, 1982, 19913, 199871, 1995546, 19967891, 199695593, 1996670090, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1996... . - Amiram Eldar, Dec 29 2024

			m = 45 - 3*3*5 and sigma(45) = 78 < 2m = 90.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Intersection of A013929 and A005100.

Cf. A087245, A087246, A087247, A008683, A000203, A005117, A005101.

Select[Range@ 280, And[! SquareFreeQ@ #, DivisorSigma[1, #] < 2 #] &] (* Michael De Vlieger, Mar 25 2017 *)

isok(n) = !issquarefree(n) && (sigma(n) < 2*n); \\ Michel Marcus, Dec 18 2013

from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
print([n for n in range(1,301) if core(n) != n and divisor_sigma(n)<2*n]) # Indranil Ghosh, Mar 26 2017

A087246 Squarefree deficient numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 110, 111, 113, 115, 118, 119

Offset: 1

Labos Elemer, Sep 09 2003

nonn

			All primes are here.
Numerous composite solutions:See A087247.
n=21=3.7 and sigma[21]=1+3+7+21=32<2.21=42.

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

Cf. A000203, A005100, A005117, A013929, A087244-A087247, A008683.

Select[Range[120], SquareFreeQ[#] && DivisorSigma[1, #] < 2 # &] (* Amiram Eldar, Aug 03 2019 *)

A087245 Nonsquarefree abundant numbers.

Original entry on oeis.org

12, 18, 20, 24, 36, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 176, 180, 192, 196, 198, 200, 204, 208, 216, 220, 224, 228, 234, 240, 252, 260, 264, 270, 272, 276, 280, 288, 294, 300, 304

Offset: 1

Labos Elemer, Sep 05 2003

nonn

			m = 20 = 2*2*5 and sigma(20) = 42 > 2m = 40.

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

Cf. A087244-A087247, A008683, A000203, A013929, A005117, A005100, A005101.

aQ[n_] := !SquareFreeQ[n] && DivisorSigma[1, n] > 2n; Select[Range[300], aQ] (* Amiram Eldar, Sep 21 2019 *)

isok(n) = !issquarefree(n) && (sigma(n) > 2*n); \\ Michel Marcus, Sep 21 2019

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A087248 Squarefree abundant numbers.

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A087244 Nonsquarefree deficient numbers.

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A087246 Squarefree deficient numbers.

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A087245 Nonsquarefree abundant numbers.

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