A087244 -id:A087244 - 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.

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 *)

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

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

A115405 Numbers n such that n^k is deficient for all k>0.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125

Offset: 1

Sergio Pimentel, Mar 08 2006

nonn

Formerly called colossally deficient numbers, but this is not a good name.
This sequence includes, but is not limited to, all prime numbers and powers of prime numbers. The only even numbers in this sequence are the powers of 2. The first odd number not in this sequence is 105. 105 is deficient but 105^2 (11025) is not. The first deficient number not in this sequence is 10.
Laatsch shows that if a number n has prime factors p1, p2,..., then the least upper bound of the sequence sigma(n^k)/n^k is p1/(p1-1) p2/(p2-1)... This equals n/phi(n), where phi is Euler's totient function. Hence n is in this sequence if 2 phi(n) >= n, which is the complement of A054741. - T. D. Noe, May 08 2006

			Let x be a deficient number (A005100, sigma(n) < 2n). Then x is colossally deficient if for every integer k > 0, x^k is also deficient.
E.g. 3 is in the sequence because 3 is deficient and also are the powers of 3 (9, 27, 81...) 22 is not in the sequence even though 22 is deficient since 22^3 = 10648 is abundant

G. C. Greubel, Table of n, a(n) for n = 1..1000
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Eric W. Weisstein's World of Mathematics, Deficient.

Cf. A005100, A005101, A087244, A000203, A083254, A089684. Complement: A054741.

fQ[n_] := Block[{k = 1}, While[k < 100 && DivisorSigma[1, n^k] < 2n^k, k++ ]; If[k == 100, True, False]]; Select[Range@ 126, fQ@ # &] (* Robert G. Wilson v, May 01 2006 *)
Select[Range[200], 2*EulerPhi[ # ]>=#&] (* T. D. Noe, May 08 2006 *)

is(n)=2*eulerphi(n)>=n \\ Charles R Greathouse IV, May 30 2013

More terms from Robert G. Wilson v, May 01 2006

Better description from T. D. Noe, May 08 2006

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A087248 Squarefree abundant numbers.

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A087246 Squarefree deficient numbers.

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A087247 Squarefree deficient nonprime numbers.

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A087245 Nonsquarefree abundant numbers.

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A115405 Numbers n such that n^k is deficient for all k>0.

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