A364977 -id:A364977 - cp's OEIS frontend

A364976 3-abundant numbers k such that k/(sigma(k)-3*k) is an integer.

180, 240, 360, 420, 540, 600, 780, 1080, 1344, 1872, 1890, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3744, 4284, 4320, 4680, 5292, 5376, 5796, 6048, 6552, 7128, 7440, 8190, 10416, 13776, 14850, 18600, 19824, 19872, 20496, 21528, 22932, 25056, 26208, 26496, 26784

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

Analogous to A153501 as 3-abundant numbers (A068403) are analogous to abundant numbers (A005101).

Numbers k such that the sum of the divisors of k except for one of them is equal to 3*k.

			180 is a term since sigma(180) - 3*180 = 6 > 0 and 180 is divisible by 6.

Amiram Eldar, Table of n, a(n) for n = 1..576 (terms below 2.5*10^10)

Subsequence of A068403.
A027687 is a subsequence.
Cf. A000203 (sigma), A005101, A005820, A055153, A153501, A160320, A218404, A218406, A329189, A364977.

Select[Range[27000], (d = DivisorSigma[1, #] - 3*#) > 0 && Divisible[#, d] &]

is(n) = {my(d = sigma(n) - 3*n); d > 0 && n%d == 0;}

A385462 Numbers t which have a proper divisor d_i(t) such that (d_i(t) + sigma(t))/t is an integer k.

Original entry on oeis.org

2, 4, 8, 10, 16, 24, 32, 44, 60, 64, 84, 128, 136, 152, 168, 184, 252, 256, 270, 336, 512, 630, 752, 756, 792, 864, 884, 924, 936, 1024, 1140, 1170, 1488, 1638, 2048, 2144, 2268, 2272, 2528, 2808, 2970, 3672, 4096, 4320, 4464, 4680, 5148, 5472, 6804, 7308, 7644, 8192, 8384

Offset: 1

Lechoslaw Ratajczak, Jun 29 2025

nonn

Consecutive elements of this sequence for which k = 2 are consecutive deficient-perfect numbers (A271816) > 1.
Consecutive elements of this sequence for which k = 3 are consecutive non-perfect elements of A364977.
Let b_k(m) be the number of elements of this sequence with the same k and <= m.
--------------------------------------------
   m   | b_2(m) | b_3(m) | b_4(m) | b_5(m) |
--------------------------------------------
 10^3  |   16   |   13   |    -   |    -   |
 10^4  |   24   |   31   |    2   |    -   |
 10^5  |   37   |   62   |    5   |    -   |
 10^6  |   54   |  107   |   19   |    -   |
 10^7  |   73   |  175   |   43   |    1   |
 10^8  |   98   |  254   |   80   |    3   |
 10^9  |  128   |  357   |  141   |   13   |
--------------------------------------------
Are there any odd terms in this sequence for which k > 2? If they exist, they are > 10^9.
Contains 2^k * (2^(k+1) + 2^j - 1) if 0 <= j <= k and 2^(k+1) + 2^j - 1 is prime. - Robert Israel, Jun 30 2025

			4 is in this sequence because sigma(4) + d_1(4) = 7 + 1 = 8 and 8/4 = 2.
24 is in this sequence because sigma(24) + d_7(24) = 60 + 12 = 72 and 72/24 = 3.
4320 is in this sequence because sigma(4320) + d_47(4320) = 15120 + 2160 = 17280 and 17280/4320 = 4.

Cf. A000005, A000203, A007691, A054027, A271816, A364977.

filter:= proc(n) local s;
  s:= - numtheory:-sigma(n) mod n;
  ormap(d -> d mod n = s, numtheory:-divisors(n) minus {n})
end proc:
select(filter, [$1..10^4]); # Robert Israel, Jun 30 2025

Select[Range[8384],AnyTrue[(Drop[Divisors[#],-1]+DivisorSigma[1,#])/#,IntegerQ]&] (* James C. McMahon, Jul 05 2025 *)

(n:1, for t:1 thru 10000 do (s:divsum(t), (A:args(divisors(t)),
              for i:1 thru length(A)-1 do (y:s+A[i],
                      if mod(y,t)=0 then (print(n,"",t), n:n+1)))));

isok(t) = my(s=sigma(t)); fordiv(t, d, if ((dMichel Marcus, Jun 30 2025

A386213 Integers t having at least one nonempty subset of the set of its proper divisors for which the equation sigma(t) + r = m*t (m is any integer > 1, r is the sum of elements of such subset) is true.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 15, 16, 18, 20, 21, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 70, 72, 75, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 112, 114, 117, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 153, 154, 156, 160, 162, 165

Offset: 1

Lechoslaw Ratajczak, Aug 12 2025

The following table lists sequences which give k-deficient-m-perfect numbers:
------------------------------------------------------------
 k/m   |     any m     |       2       |         3         |
------------------------------------------------------------
 any k | this sequence | A331627 \ {1} |         -         |
------------------------------------------------------------
 1     | A385462       | A271816 \ {1} | A364977 \ A000396 |
------------------------------------------------------------
 2     |       -       | A331628       |         -         |
------------------------------------------------------------
 3     |       -       | A331629       |         -         |
------------------------------------------------------------
This sequence contains all, and only, (any k)-deficient-m-perfect numbers (m = 2,3,4,...), equivalently it contains all, and only, k-deficient-(any m)-perfect numbers (k = 1,2,3,...).

			24 is a term because for 24 the set of proper divisors is {1, 2, 3, 4, 6, 8, 12} and it has exactly 6 subsets which sum up to r satisfying the equation sigma(24) + r = k*24:
  (1) sigma(24) + d_7(24) = 60 + 12 = 72 and 72 = 3*24,
  (2) sigma(24) + (d_4(24) + d_6(24)) = 60 + (4 + 8) = 72 and 72 = 3*24,
  (3) sigma(24) + (d_2(24) + d_4(24) + d_5(24)) = 60 + (2 + 4 + 6) = 72 and 72 = 3*24,
  (4) sigma(24) + (d_1(24) + d_3(24) + d_6(24)) = 60 + (1 + 3 + 8) = 72 and 72 = 3*24,
  (5) sigma(24) + (d_1(24) + d_2(24) + d_3(24) + d_5(24)) = 60 + (1 + 2 + 3 + 6) = 72 and 72 = 3*24,
  (6) sigma(24) + (d_1(24) + d_2(24) + d_3(24) + d_4(24) + d_5(24) + d_6(24) + d_7(24)) = 60 + (1 + 2 + 3 + 4 + 6 + 8 + 12) = 96 and 96 = 4*24.
So 24 is (1, 2, 3 (in 2 variants), 4)-deficient-3-perfect and 7-deficient-4-perfect number.

Cf. A000005, A000203, A007691, A331627, A385462.

n = 1;l={};Do[x = 1;s=DivisorSigma[1,t];A=Most[Divisors[t]];B=Subsets[A];  Do[r=Total[B[[i]]];If[Mod[s+r,t]==0,x=x+1],{i,2,2^Length[A]}];  If[x>1,AppendTo[l,t];n=n+1],{t,1,165}];l (* James C. McMahon, Aug 25 2025 *)

(n:1, for t:1 thru 300 do (x:1, s:divsum(t), A:delete(t, divisors(t)), B:args(powerset(A)),
              for i:2 thru 2^(length(args(A))) do (r:apply("+", args(B[i])),
                      if mod(s+r, t)=0 then (x:x+1)),
                                       if x>1 then (print(n, "", t), n:n+1)));

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A364976 3-abundant numbers k such that k/(sigma(k)-3*k) is an integer.

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A385462 Numbers t which have a proper divisor d_i(t) such that (d_i(t) + sigma(t))/t is an integer k.

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A386213 Integers t having at least one nonempty subset of the set of its proper divisors for which the equation sigma(t) + r = m*t (m is any integer > 1, r is the sum of elements of such subset) is true.

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