A023198 -id:A023198 - cp's OEIS frontend

A103288 Numbers k such that sigma(k) >= 2k-1 (union of perfect, abundant and least deficient numbers).

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 128, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220

Offset: 1

Max Alekseyev, Jan 28 2005

nonn

If the only least deficient numbers are the powers of 2 (open problem) then this sequence is the union of A023196 and A000079.

Like the abundant numbers, this sequence has density between 0.2474 and 0.2480, see A005101. - Charles R Greathouse IV, Nov 30 2022

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A023196, A023197, A023198, A023199, A000079, A004125.

Select[Range[500], DivisorSigma[1, #] >= 2*# - 1 &] (* Paolo Xausa, Dec 09 2024 *)

for(n=1,1000,if(sigma(n)>=2*n-1,print(n)));

Numbers k such that A004125(k) <= A004125(k-1).

A204831 Numbers n whose divisors can be partitioned into four disjoint sets whose sums are all sigma(n)/4.

Original entry on oeis.org

27720, 30240, 32760, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 143640, 151200, 163800, 166320, 171360, 180180, 181440, 184800, 191520

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

Subsequence of A023198 (numbers n such that sigma(n) >= 4n).

			Number 27720 is in the sequence because sigma(27720)/4 = 28080 = 360 + 27720 = 20 + 60 + 280 + 2310 + 4620 + 6930 + 13860 = 9 + 30 + 420 + 1540 + 1980 + 2772 + 3080 + 3465 + 5544 + 9240 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 10 + 11 + 12 + 14 + 15 + 18 + 21 + 22 + 24 + 28 + 33 + 35 + 36 + 40 + 42 + 44 + 45 + 55 + 56 + 63 + 66 + 70+ 72 + 77 + 84 + 88 + 90 + 99 + 105 + 110 + 120 + 126 + 132 + 140 + 154 + 165 + 168 + 180 + 198 + 210 + 220 + 231 + 252 + 264+ 308 + 315 + 330 + 385 + 396 + 440 + 462 + 495 + 504 + 616 + 630 + 660 + 693 + 770 + 792 + 840 + 924 + 990 + 1155 + 1260 + 1320 + 1386 + 1848 + 2520 + 3960 (summands are all divisors of 27720).

Farid Jokar, On k-layered numbers, arXiv:2207.09053 [math.NT], 2022.

Cf. A083207 (Zumkeller numbers--numbers n whose divisors can be partitioned into two disjoint sets whose sums are both sigma(n)/2), A204830 (numbers n whose divisors can be partitioned into three disjoint sets whose sums are all sigma(n)/3).

with(numtheory);with(combstruct);
A204831:=proc(i)
local S,R,Stop,Comb,c,d,k,m,n,s;
for n from 1 to i do
  s:=sigma(n); c:=op(divisors(n));
  if (modp(s,4)=0 and 4*n<=s) then
     S:=1/4*s-n; R:=select(m->m<=S,[c]); Stop:=false;
     Comb:=iterstructs(Combination(R));
     while not (finished(Comb) or Stop) do
       Stop:=add(d,d=nextstruct(Comb))=S;
     od;
     if Stop then print(n); fi;
  fi;
od;
end:
A204831(100000); # Paolo P. Lava, Jan 24 2012

A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

902880, 1534680, 361674720, 767685600, 4530770640, 4941414720, 5405788800, 5517818880, 16993944000, 20429240832, 94820077440

Offset: 1

Amiram Eldar, Oct 01 2022

Analogous to 4-perfect numbers (A027687) with nonsquarefree divisors.
Equivalently, numbers k such that A325314(k) = -3*k.
There are no numbers k below 10^11 such that A162296(k) = m*k for integers m > 4.

			902880 is a term since A162296(902880) = 3611520 = 4*902880.

Cf. A162296, A325314.
Subsequence of A013929 and A023198.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), A322609 (m=2), A357493 (m=3), this sequence (m=4).
Similar sequence: A027687.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) == 4*n]; Select[Range[2, 2*10^6], q]

A204828 Numbers n with abundancy 3 <= sigma(n)/n < 4.

Original entry on oeis.org

120, 180, 240, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1080, 1200, 1260, 1320, 1344, 1440, 1512, 1560, 1584, 1620, 1680, 1800, 1848, 1872, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2280, 2340, 2352, 2376, 2400, 2520, 2640

Offset: 1

Jaroslav Krizek, Jan 22 2012

nonn

A subsequence of A023197 (numbers n such that sigma(n) >= 3n) which is in turn a subsequence of the abundant numbers A005101, i.e., numbers n with sigma(n)/n > 2.
Differs from A023197 from a(565) on: The first term of A023197 which is not in this sequence is A023197(565) = 27720 = A023198(1) = A023199(4), the least number with abundancy >= 4.
Numbers with abundancy sigma(n)/n < 2 are called deficient and listed in A005100. Numbers with sigma(n)/n in the interval [2,3) are listed in A204829. Numbers with sigma(n)/n in the interval [4,5) are listed in A230608. - M. F. Hasler, Dec 05 2013

			Number 180 is in the sequence because sigma(180)/180 = 546/180 = 3.0333...

Jaroslav Krizek, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A204829 (abundant numbers with abundancy 2 <= a < 3).

A069057 Numbers k such that the numerator of Sum_{d|k} 1/d > 2k.

Original entry on oeis.org

36, 100, 144, 324, 392, 400, 576, 784, 800, 900, 968, 1296, 1352, 1575, 1600, 1936, 2304, 2450, 2500, 2704, 2916, 3136, 3600, 3872, 4356, 4624, 5184, 5408, 5776, 6050, 6272, 6400, 7744, 8100, 8450, 8464, 9216, 9248, 10000, 10404, 10816, 11552, 11664

Offset: 1

Benoit Cloitre, Apr 07 2002

Many terms in the sequence are squares: 6^2,10^2,12^2,18^2,20^2,24^2,28^2,30^2,36^2,40^2,44^2,48^2,50^2...

Abundant numbers that are coprime to their abundance (intersection of A005101 and A014567) are all terms. Abundant numbers that are not coprime to their abundance are 4-abundant (i.e., terms of A023198, e.g., 64468800, 121726800, 168480000, ...). - Waldemar Puszkarz and Amiram Eldar, Aug 15 2024

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

Cf. A005101, A014567, A017665, A023198.

Select[Range[12000], Numerator[Total[1/Divisors@ #]] > 2 # &] (* Michael De Vlieger, May 04 2016 *)

isok(n) = numerator(sigma(n)/n) > 2*n; \\ Michel Marcus, May 04 2016

A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

Original entry on oeis.org

27720, 30240, 32760, 50400, 55440, 60480, 65520, 75600, 83160, 85680, 90720, 95760, 98280, 100800, 105840, 110880, 115920, 120120, 120960, 128520, 131040, 138600, 141120, 143640, 151200, 163800, 166320, 171360, 176400, 180180, 181440, 184800, 191520, 194040

Offset: 1

Jaroslav Krizek, Nov 29 2013

nonn

A subsequence of A023198 (numbers with abundancy >= 4). It differs from A023198 from a(31093) on: The term A023198(31093) = 122522400 = A023199(5) = A215264(1) is not in this sequence. It excludes all terms of A215264, but also the 5-perfect numbers A046060, which are neither in this sequence nor in A215264. [Corrected by M. F. Hasler, Dec 05 2013]
A108775(a(n)) = 4.
There are 31092 terms less than 122522399. - T. D. Noe, Dec 04 2013

			27720 is in sequence because sigma(27720) / 27720 = 112320 / 27720 = 4.0519....

Jaroslav Krizek, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Abundancy
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A000203, A023198, A023199, A108775.
Cf. A005100 (deficient numbers with abundancy 1 <= a < 2),
Cf. A204829 (numbers with abundancy 2 <= a < 3),
Cf. A204828 (abundant numbers with abundancy 3 <= a < 4).
Cf. A215264 (abundant numbers with abundancy > 5).

Select[Range[200000], 4 <= DivisorSigma[1, #]/# < 5 &] (* T. D. Noe, Dec 04 2013 *)

Corrected and edited by M. F. Hasler, Dec 05 2013

A291458 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 4.

Original entry on oeis.org

27720, 60480, 65520, 90720, 98280, 105840, 115920, 120120, 120960, 128520, 131040, 143640, 151200, 163800, 180180, 191520, 205920, 207900, 211680, 218400, 229320, 235620, 241920, 249480, 264600, 272160, 289800, 292320, 312480, 332640, 360360, 372960, 393120, 414960

Offset: 1

Paolo P. Lava, Aug 24 2017

Case k=2 are the admirable numbers (A111592).

Subset of A023198, A068404, A204831, A230608.

			One of the proper divisors of 27720 is 360 and sigma(27720) - 4*360 = 112320 - 1440 = 110880 = 4*27720.
One of the proper divisors of 115920 is 144 and sigma(115920) - 4*144 = 464256 - 576 = 463680 = 4*115920.

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

Cf. A000203, A023198, A068404, A111592, A204831, A230608, A291457, A291459.

with(numtheory): P:=proc(q,h) local a,k,n; for n from 1 to q do a:=sort([op(divisors(n))]);
for k from 1 to nops(a)-1 do if sigma(n)-h*a[k]=h*n then print(n); break; fi; od; od; end: P(10^9,4);

With[{k = 4}, Select[Range[5 * 10^5], Function[n, AnyTrue[Most@ Divisors@ n, DivisorSigma[1, n] - k # == k n &]]]] (* Michael De Vlieger, Aug 24 2017 *)
(* or *)
k=4; Select[Range[5*^5], (t = DivisorSigma[1, #]/k - #; #>t>0 && IntegerQ[t] && Mod[#, t] == 0) &] (* much faster, Giovanni Resta, Aug 25 2017 *)

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A103288 Numbers k such that sigma(k) >= 2k-1 (union of perfect, abundant and least deficient numbers).

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A204831 Numbers n whose divisors can be partitioned into four disjoint sets whose sums are all sigma(n)/4.

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A357494 Numbers k such that s(k) = 4*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A204828 Numbers n with abundancy 3 <= sigma(n)/n < 4.

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A069057 Numbers k such that the numerator of Sum_{d|k} 1/d > 2k.

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A230608 Numbers with abundancy 4 <= sigma(n)/n < 5.

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A291458 Numbers n having a proper divisor d such that sigma(n) - k*d = k*n. Case k = 4.

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A291458 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 4.