cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Lechoslaw Ratajczak, Jul 18 2025

Keywords

Comments

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.

Examples

			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).

Crossrefs

Cf. A000203, A271816, A331627.

Programs

  • Maxima
    (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))));
  • PARI
    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

A331628 Integers that are exactly 2-deficient-perfect numbers.

Original entry on oeis.org

15, 21, 45, 50, 52, 63, 75, 99, 105, 117, 135, 182, 190, 195, 230, 231, 266, 273, 315, 375, 405, 435, 495, 585, 592, 656, 688, 850, 891, 950, 1155, 1215, 1305, 1365, 1395, 1612, 1755, 1845, 1862, 1875, 1892, 1989, 2079, 2295, 2312, 2332, 2336, 2350, 2366, 2475
Offset: 1

Views

Author

Michel Marcus, Jan 23 2020

Keywords

Comments

Numbers k that have 2 distinct proper divisors, d_1 and d_2, such that sigma(k) = 2*k - (d_1 + d_2). - Amiram Eldar, Dec 29 2024

Examples

			117 is an exactly 2-deficient-perfect number with d1=13 and d2=39: sigma(117) = 182 = 2*117 - (13 + 39). See Theorem 1 p. 2 of FengJuan Chen.

Links

Crossrefs

Cf. A000203 (sigma), A271816 (deficient-perfect numbers (k=1)), A331627 (k-deficient-perfect), A331629 (3-deficient-perfect).

Programs

  • Mathematica
    def2[n_] := Catch@Block[{s = 2*n - DivisorSigma[1, n], d}, If[s > 0, d = Most@ Divisors@ n; Do[If[s == d[[i]] + d[[j]], Throw@ True], {i, 2, Length@ d}, {j, i-1}]; False]]; Select[Range[2500], def2] (* Giovanni Resta, Jan 23 2020 *)

Extensions

More terms from Giovanni Resta, Jan 23 2020

A331629 Integers that are exactly 3-deficient-perfect numbers.

Original entry on oeis.org

130, 154, 170, 182, 232, 250, 290, 434, 484, 848, 944, 950, 988, 1196, 1210, 1274, 1276, 1450, 1521, 1564, 1666, 1892, 1924, 2618, 2848, 2888, 2926, 3094, 3232, 3424, 3458, 3542, 3616, 4186, 4214, 4250, 4522, 4750, 4810, 5150, 5278, 5330, 5510, 5590, 5642, 5890
Offset: 1

Views

Author

Michel Marcus, Jan 23 2020

Keywords

Comments

Aursukaree & Pongsriiam prove that 1521 is the only odd term with at most two distinct prime factors.
Numbers k that have 3 distinct proper divisors, d_1, d_2 and d_3, such that sigma(k) = 2*k - (d_1 + d_2 + d_3). - Amiram Eldar, Dec 29 2024

Examples

			130 is an exactly 3-deficient-perfect number with d1=1, d2=2 and d3=5: sigma(130) = 252 = 2*130 - (1+2+5).

Links

Crossrefs

Cf. A000203 (sigma), A271816 (deficient-perfect numbers (k=1)), A331627 (k-deficient-perfect), A331628 (2-deficient-perfect).

Programs

  • Mathematica
    def3[n_] := Catch@ Block[{s = 2*n - DivisorSigma[1, n], d}, If[s > 0, d = Most@ Divisors@ n; Do[If[s == d[[i]] + d[[j]] + d[[k]], Throw@ True], {i, 3, Length@ d}, {j, i-1}, {k, j-1}]; False]]; Select[ Range[6000], def3] (* Giovanni Resta, Jan 23 2020 *)

Extensions

More terms from Giovanni Resta, Jan 23 2020

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

Views

Author

Lechoslaw Ratajczak, Aug 12 2025

Keywords

Comments

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,...).

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • Maxima
    (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)));
Showing 1-4 of 4 results.

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