A271816 -id:A271816 - cp's OEIS frontend

A386317 Integers t which satisfy 3/2 <= abundancy(t) < 2 but which are not k-deficient-perfect numbers A331627.

14, 22, 26, 34, 38, 46, 58, 62, 68, 74, 76, 82, 86, 92, 94, 98, 106, 110, 116, 118, 122, 124, 134, 142, 146, 147, 148, 158, 164, 166, 171, 172, 178, 188, 194, 202, 206, 212, 214, 218, 225, 226, 236, 242, 244, 248, 254, 255, 262, 268, 274, 278, 284, 285, 286, 292, 296, 298, 302, 314

Offset: 1

Lechoslaw Ratajczak, Jul 18 2025

nonn

A necessary (but not sufficient) condition for an integer t to be a k-deficient-m-perfect number: (m + 1)/2 <= abundancy(t) < m:
- for m = 2: 3/2 <= abundancy(t) < 2,
- for m = 3: 2 <= abundancy(t) < 3,
- for m = 4: 5/2 <= abundancy(t) < 4.

			13 is not in this sequence because abundancy(13) = 14/13 (14/13 < 3/2).
14 is in this sequence because abundancy(14) = 12/7 (3/2 <= 12/7 < 2) but 14 is not a k-deficient-perfect number (therefore is not included in A331627).
15 is not in this sequence because abundancy(15) = 8/5 (3/2 <= 8/5 < 2) but 15 is a k-deficient-perfect number (therefore is included in A331627).

Cf. A000203, A271816, A331627.

(n:1, abundancy(x):=divsum(x)/x,
     for t:1 thru 500 do
        (if abundancy(t)>=3/2 and abundancy(t)<2 then
        (A:append(args(powerset(delete(t,divisors(t)))),[{0}]), b:length(A),
            for i:1 unless (divsum(t)+apply("+" , args(A[i])))/t=2 or i>=b do j:i,
               if j>=b-1 then (print(n , "" , t), n:n+1))));

isok(m) = my(d=divisors(m), ss=vecsum(d), ab=sigma(m)/m); if ((ab>=3/2) && (ab<2), d = Vec(d, #d-1); forsubset(#d, s, if (#s && (sum(i=1, #s, d[s[i]]) == 2*m - ss), return(0))); return(1)); \\ Michel Marcus, Jul 19 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)));

cp's OEIS Frontend

A386317 Integers t which satisfy 3/2 <= abundancy(t) < 2 but which are not k-deficient-perfect numbers A331627.

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