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 1-10 of 12 results. Next

A097498 Numbers k divisible by their abundance sigma(k) - 2*k.

Original entry on oeis.org

1, 2, 4, 8, 10, 12, 16, 18, 20, 24, 32, 40, 44, 56, 64, 88, 104, 120, 128, 136, 152, 184, 196, 224, 234, 256, 368, 464, 512, 650, 672, 752, 884, 992, 1024, 1504, 1888, 1952, 2048, 2144, 2272, 2528, 3724, 4096, 5624, 8192, 8384, 9112, 11096, 12224, 13736
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 24 2004

Keywords

Links

Crossrefs

Cf. A000079 (subsequence), A033879, A033880.
Disjoint union of A153501 and A271816.
Cf. also A379236.

Programs

  • Mathematica
    Select[Range[15000],Divisible[#,DivisorSigma[1,#]-2#]&]//Quiet (* Harvey P. Dale, Mar 15 2018 *)
  • PARI
    is(n)=my(t=sigma(n)-2*n); t && n%t==0 \\ Charles R Greathouse IV, Dec 12 2014

A331627 Integers that are (exactly) k-deficient-perfect numbers.

Original entry on oeis.org

1, 2, 4, 8, 10, 15, 16, 21, 32, 44, 45, 50, 52, 63, 64, 75, 99, 105, 117, 128, 130, 135, 136, 152, 153, 154, 165, 170, 182, 184, 189, 190, 195, 207, 230, 231, 232, 238, 250, 256, 266, 273, 290, 297, 310, 315, 322, 351, 375, 399, 405, 429, 434, 435, 441, 459, 484, 495, 512
Offset: 1

Views

Author

Michel Marcus, Jan 23 2020

Keywords

Comments

An integer m is an exactly k-deficient-perfect number if it has k distinct proper divisors d_i such that sigma(m) = 2*m - Sum_{i=1..k} d_i.

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 Chen.

Links

Crossrefs

Cf. A000203 (sigma), A271816 (deficient-perfect numbers (k=1)), A331628 (2-deficient-perfect), A331629 (3-deficient-perfect), A386317.

Programs

  • Mathematica
    kdef[n_] := n == 1 || Block[{s = 2*n - DivisorSigma[1, n], d}, If[s <= 0, False, d = Most@ Divisors@ n; MemberQ[ Total /@ Subsets[d, {1, Length@ d}], s]]]; Select[ Range[512], kdef] (* Giovanni Resta, Jan 23 2020 *)
  • PARI
    isok(m) = my(d=divisors(m), ss=sigma(m)); d = Vec(d, #d-1); forsubset(#d, s, if (#s && (sum(i=1, #s, d[s[i]]) == 2*m - ss), return(1));); \\ Michel Marcus, Dec 29 2024

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

A364977 Numbers k such that k/(3*k - sigma(k)) is a positive integer.

Original entry on oeis.org

6, 24, 28, 60, 84, 168, 252, 270, 336, 496, 630, 756, 792, 864, 924, 936, 1140, 1170, 1488, 1638, 2268, 2808, 2970, 3672, 4464, 5148, 5472, 6804, 7308, 7644, 8128, 8700, 8910, 9300, 9936, 11172, 13392, 16368, 18018, 20196, 20412, 22230, 24384, 25116, 27888, 31968
Offset: 1

Views

Author

Amiram Eldar, Aug 15 2023

Keywords

Comments

Analogous to A271816 as 3-abundant numbers (A068403) are analogous to abundant numbers (A005101).
Numbers k such that the sum of the divisors of k with one of them added twice is equal to 3*k.
The perfect numbers (A000396) are all terms.
For all the terms k, 2 <= sigma(k)/k < 3, i.e., they are all nondeficient numbers (A023196), and none are 3-abundant (A068403).

Examples

			6 is a term since 3*6 - sigma(6) = 6 > 0 and 6 is divisible by 6.
24 is a term since 3*24 - sigma(24) = 12 > 0 and 24 is divisible by 12.

Links

Crossrefs

Cf. A000203 (sigma), A000396, A023196, A068403, A271816, A329189, A364976.

Programs

  • Mathematica
    Select[Range[32000], (d = 3*# - DivisorSigma[1, #]) > 0 && Divisible[#, d] &]
  • PARI
    is(n) = {my(d = 3*n - sigma(n)); d > 0 && n%d == 0;}

A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].

Original entry on oeis.org

10, 12, 18, 20, 24, 40, 56, 88, 104, 116, 136, 184, 196, 224, 312, 368, 428, 464, 520, 528, 650, 672, 760, 884, 992, 1472, 1504, 1888, 1952, 2528, 3424, 3724, 4832, 5312, 6464, 7136, 9112, 11096, 11288, 11744, 13216, 15352, 15376, 15872, 15968, 16256, 17816, 17964, 22616, 24448, 26728, 28544, 29296, 30592, 30656
Offset: 1

Views

Author

Antti Karttunen, Jan 05 2025

Keywords

Comments

Among the first 484 terms, there are no odd numbers, the only squares are 196, 15376, 1032256, and 18 is the only twice square.

Examples

			196 is a term as sigma(196) = 399, 2*196 XOR 399 = 7 is not zero, and A048720(7, 89) = 399.

Links

Crossrefs

Cf. A000203, A000396, A003987, A048720, A280500, A318467.
Cf. A379234 (subsequence).
Cf. also A097498 (= A153501 U A271816).

Programs

  • PARI
    divides_in_GF2X(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); !lift(Pa % Pb); };
is_A379236(n) = { my(s=sigma(n), x=bitxor(2*n, s)); (x && divides_in_GF2X(n, x)); };

Formula

{k such that k = A048720(A318467(k), x) for some x > 0}.
{k not in A000396 such that A280500(k, A318467(k)) > 0}.

A303358 Bi-unitary deficient-perfect numbers: bi-unitary deficient numbers k for such that 2*k - bsigma(k) is a bi-unitary divisor of k, where bsigma(k) is the sum of bi-unitary divisors of k (A188999).

Original entry on oeis.org

1, 2, 8, 10, 12, 32, 112, 128, 136, 144, 152, 184, 512, 1088, 2048, 2144, 2272, 2528, 2736, 3248, 3312, 4592, 7936, 8192, 9800, 11800, 17176, 18632, 18904, 22984, 32768, 32896, 33664, 34688, 49024, 57152, 77248, 85952, 131072, 176400, 212400, 309168, 335376
Offset: 1

Views

Author

Amiram Eldar and Michael De Vlieger, Apr 22 2018

Keywords

Comments

The bi-unitary version of A271816.
Includes all the odd powers of 2 (A004171).

Examples

			112 is in the sequence since the sum of its bi-unitary divisors is 1 + 2 + 7 + 8 + 14 + 16 + 56 + 112 = 216 and 2*112 - 216 = 8 is a bi-unitary divisor of 112.

Links

Crossrefs

Cf. A004171, A188999, A271816, A292982, A303359.

Programs

  • Mathematica
    f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; biunitaryDivisorQ[ div_, n_] := If[Mod[#2,#1]==0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]]&, {#1, #2/#1}]]==1, False]& @@{div, n}; aQ[n_] := Module[{d=2n-bsigma[n]},If[d<=0, False,biunitaryDivisorQ[d,n]]]; s={}; Do[ If[aQ[n], AppendTo[s,n]], {n, 1, 10000}]; s
  • PARI
    udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
isok(n) = my(divs = biudivs(n), sig = vecsum(divs)); (sig < 2*n) && vecsearch(divs, 2*n-sig); \\ Michel Marcus, Apr 27 2018

A305617 Deficient 2-hyperperfect numbers: numbers k such that 3*k/2 + 1/2 - sigma(k) is a proper divisor of k.

Original entry on oeis.org

3, 9, 27, 35, 39, 55, 81, 243, 279, 387, 715, 729, 1443, 2187, 2619, 3655, 5635, 6561, 10855, 12635, 19683, 59049, 77283, 177147, 178119, 294759, 443135, 531441, 817167, 1170723, 1594323, 1605987, 1632231, 1710963, 1947159, 2410239, 2624375, 2655747, 3944255
Offset: 1

Views

Author

Amiram Eldar, Jun 06 2018

Keywords

Comments

Includes all the powers of 3 (A000244).
A combination of the notions 2-hyperperfect numbers (A007593) and deficient-perfect numbers (A271816).

Examples

			35 is in the sequence since sigma(35) = 48 and 3*35/2 + 1/2 - 48 = 5 is a proper divisor of 35.

Links

Crossrefs

Cf. A000203, A000244, A007593, A271816, A305616.

Programs

  • Mathematica
    aQ[n_] := Module[{d = 3n/2+1/2-DivisorSigma[1,n]}, d>0 && d!=n && IntegerQ[d] && Divisible[n,d]]; Select[Range[2,1000000], aQ]
  • PARI
    isok(n) = (n % 2) && (k = (3*n+1)/2 - sigma(n)) && (k > 0) && !(n % k) && (k != n); \\ Michel Marcus, Jun 07 2018, corrected by Amiram Eldar, Dec 23 2024

A303356 Unitary deficient-perfect numbers: unitary deficient numbers k such that 2*k-usigma(k) is a unitary divisor of k, where usigma is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

1, 2, 10, 12, 120, 4080, 5280, 6720, 17472, 137280, 174720, 908160, 29621760, 31100160, 41879040, 89806080, 99240960, 101391360, 143969280, 226652160, 466794240, 732103680, 760488960, 779412480, 916016640, 918382080, 951498240, 1001172480, 1365450240, 3151948800, 9464663040
Offset: 1

Views

Author

Amiram Eldar, Apr 22 2018

Keywords

Comments

The unitary version of A271816.

Examples

			120 is in the sequence since 2*120 - usigma(120) = 240 - 216 = 24, and 24 is a unitary divisor of 120.

Crossrefs

Cf. A034448, A271816, A303357.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; aQ[n_] := Module[{d}, d=2n-usigma[n]; If[ d<=0, False, Divisible[n,d] && GCD[d, n/d] == 1 ]]; Select[Range[100000], aQ]

Extensions

a(19)-a(31) from Giovanni Resta, Apr 26 2018

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

Views

Author

Lechoslaw Ratajczak, Jun 29 2025

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    Select[Range[8384],AnyTrue[(Drop[Divisors[#],-1]+DivisorSigma[1,#])/#,IntegerQ]&] (* James C. McMahon, Jul 05 2025 *)
  • Maxima
    (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)))));
  • PARI
    isok(t) = my(s=sigma(t)); fordiv(t, d, if ((dMichel Marcus, Jun 30 2025
Showing 1-10 of 12 results. Next

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