A129521 -id:A129521 - cp's OEIS frontend

A231814 Squarefree numbers (from A005117) with prime divisors in a 2p-1 progression.

6, 15, 30, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, 51319, 79003, 88831, 104653, 146611, 188191, 218791, 226801, 269011, 286903, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1314631, 1373653, 1537381, 1755001, 1869211

Offset: 1

Jaroslav Krizek, Nov 13 2013

nonn

Squarefree numbers with k >= 2 prime factors of the form p_1 * p_2 * ... * p_k, where p_1 < p_2 < ... < p_k = primes with p_k = 2 * p_(k-1) - 1.
Each of these numbers is divisible by the arithmetic mean of its proper divisors.
Supersequence of A129521 (numbers of the form p*q, p and q prime with q=2*p-1; see A005382 and A005383).

			51319 = 19*37*73 where 37 = 2*19 - 1, 73 = 2*37 - 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A005117, A005382, A005383, A129521, A231815, A231816.

Cf. A057330 (first prime for such numbers that has n factors).

N:= 10^7: # for terms <= N
p:= 1: S:= NULL: count:= 0:
do
  p:= nextprime(p);
  if p*(2*p-1) > N then break fi;
  q:= p; x:= p;
  do
    q:= 2*q-1;
    if not isprime(q) then break fi;
    x:= x*q;
    if x > N then break fi;
    S:= S,x; count:= count+1;
  od;
od:
sort([S]); # Robert Israel, Mar 24 2023

geomQ[lst_] := Module[{x = lst - 1}, x = x/x[[1]]; Log[2, x] + 1 == Range[Length[x]]]; Select[Range[2, 1000000], ! PrimeQ[#] && SquareFreeQ[#] && geomQ[Transpose[FactorInteger[#]][[1]]] &] (* T. D. Noe, Nov 14 2013 *)

A259676 Heptagonal numbers (A000566) that are semiprimes (A001358).

Original entry on oeis.org

34, 55, 235, 403, 469, 697, 1177, 1651, 2059, 2839, 4141, 5221, 6943, 9211, 9517, 13213, 13579, 21949, 23377, 25351, 29539, 31753, 34633, 37027, 53071, 62173, 68641, 74563, 78943, 93799, 96727, 118483, 130759, 144841, 164737, 171217, 187279, 191407, 196981

Offset: 1

Colin Barker, Jul 03 2015

nonn

For these semiprimes k*(5*k-3)/2, the corresponding k are listed in A114517.

			The heptagonal number 34 is in the sequence because 34 = 2 * 17.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A129521, A245365, A259677.

IsSemiprime:=func; [s: n in [2..300] | IsSemiprime(s) where s is n*(5*n-3) div 2]; // Vincenzo Librandi, Jul 04 2015

a={}; Do[If[PrimeOmega[n (5 n - 3) / 2]==2, AppendTo[a, n(5 n - 3) / 2]], {n, 1, 200}]; a (* Vincenzo Librandi, Jul 04 2015 *)
Select[PolygonalNumber[7,Range[300]],PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 07 2021 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
select(n->bigomega(n)==2, vector(2000, n, pg(7, n)))

Equals A000566 intersect A001358.

A335267 Composite numbers whose harmonic mean of their divisors that are larger than 1 is an integer.

Original entry on oeis.org

6, 15, 28, 30, 91, 117, 135, 252, 270, 496, 703, 864, 936, 1891, 1989, 2295, 2701, 4284, 4590, 5733, 8128, 8432, 12403, 18721, 19872, 21528, 38503, 41580, 49141, 51319, 56896, 79003, 88831, 104653, 121920, 146611, 188191, 218791, 226801, 235053, 269011, 286903

Offset: 1

Amiram Eldar, May 29 2020

nonn

The primes are excluded from this sequence since they are trivial terms.
The corresponding harmonic means are 3, 5, 5, 5, 13, 9, 9, 9, 9, 9, 37, ...
Equivalently, composite numbers m such that (sigma(m)-m) | m*(tau(m)-1), or A001065(m) | A168014(m).
The semiprimes terms of this sequence are of the form p*q where p and q = 2*p - 1 are primes (A129521).
If m is a k-perfect numbers, k = 2, 3, ... (i.e., sigma(m) = k*m), then sigma(m)-m = (k-1)*m. If (k-1)*m | m*(tau(m)-1) then (k-1) | (tau(m)-1). If k is odd then tau(m) is also odd, so m is a square, and sigma(m) is odd. Since m | sigma(m) this means that m is also odd. Since there is no known odd multiply-perfect number except for 1 (A007691), there are no known k-perfect numbers with odd k in this sequence.
The perfect numbers (k=2, A000396) are terms: if m is a perfect number then sigma(m)-m = m.
The 4-perfect number (k=4, A027687) m are terms if 3 | (tau(m)-1). Of the first 36 terms of A027687 there are 8 such terms, the first is A027687(26).
The 6-perfect number (k=6, A046061) m are terms if 5 | (tau(m)-1). Of the first 245 terms of A046061 there are 20 such terms, the first is A046061(19).
Hemiperfect numbers that are terms of this sequence include A055153(i) for i = 10, 18 and 20, A141645(21), and A159271(i) for i = 97 and 103.

			6 is a term since its divisors other than 1 are 2, 3 and 6, and their harmonic mean, 3/(1/2 + 1/3 + 1/6) = 3, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..1000

A000396 and A129521 are subsequences.
Similar sequences: A001599, A247077, A247078.
Cf. A000005 (tau), A000203 (sigma).
Cf. A001065, A032741, A168014,
Cf. A005382, A005383.
Cf. A027687, A046061, A055153, A141645, A159271.

Select[Range[10^6], CompositeQ[#] && Divisible[# * (DivisorSigma[0, #] - 1), DivisorSigma[1, #] - #] &]
Select[Range[287000],CompositeQ[#]&&IntegerQ[HarmonicMean[ Rest[ Divisors[ #]]]]&] (* Harvey P. Dale, Jan 21 2021 *)

A338488 Numbers n whose symmetric representation of sigma(n) has a maximum width of 2 that occurs exactly once (at the diagonal).

Original entry on oeis.org

6, 15, 28, 91, 153, 190, 325, 496, 703, 861, 946, 1431, 1891, 2278, 2701, 3655, 4753, 5151, 5356, 5995, 6441, 7381, 8128, 8911, 9453, 10585, 11476, 12403, 13366, 15051, 18721, 21115, 22366, 23653

Offset: 1

Hartmut F. W. Hoft, Oct 30 2020

nonn

For numbers n in the sequence the symmetric representation of sigma(n) consists of an odd number of regions. The geometric form for a single width 2 at the diagonal requires that
(a) the length of the last leg of the n-th Dyck path is A237591(n,row(n)) = 1 and the leg horizontal, which implies that row(n) is odd, and
(b) the length of the last leg of the (n-1)-st Dyck path is A237591(n-1,row(n-1)) = 2 and the leg vertical, which implies that row(n-1) = row(n) - 1 is even:
    |_
  |   |_
  | 
Therefore, n is a hexagonal number (see A000384) and row(n) is an odd divisor of n since otherwise the central region of the symmetric representation for sigma(n) defined by the largest odd divisor d < row(n) would create an extent of width 2 larger than 1.
Since all other regions must have width 1 the following conditions characterize the numbers in this sequence:
Let 1 = d_1 < d_2 < ... < d_s <= row(n) < d_(s+1) < ... < d_t = q be all odd divisors of n = 2^k * q, k >= 0. n is hexagonal, 2^(k+1) * d_i < d_(i+1), for 1 <= i <= s-2, and d_s = row(n), so that 2^(k+1) * d_(s-1) = d_s + 1.
The numbers of this sequence can be arranged as a table according to the number of regions (one fewer than the number of its odd divisors) in the symmetric representation of sigma(n):
     1      3       5       7           9        11
  -------------------------------------------------
     6     15     153     861      195625     43071
    28     91     325    1431         ...       ...
   496    190    4753    3655  6859425628  50999950
  8128    703    7381    5151         ...       ...
   ...    946     ...    5995
         1891  468028    6441
         2278     ...    8911
         2701            9453
         5356           10585
        11476           15051
        12403           21115
        13366           23653
        18721             ...
        22366          124750
          ...             ...
The numbers less than 25000 in the first four columns were computed using function a338488[] while the numbers in the remaining two columns and the first even numbers in the 5-column and 7-column were computed by a function implementing the conditions on the structure of the odd divisors.
The 1-column consists of the even perfect numbers, A000396.
The 3-column is the sequence of numbers n =2^k * p * q, p & q odd primes, such that 2^(k+1) < p < q < 2^(k+1)*p = q+1. It is a subsequence of A338486, and includes the odd numbers in A129521 since (q+1)/2 = p is prime.
The odd numbers in the 5-column have 6 divisors and therefore form a subsequence in A116565.
Conjecture: For every odd number 2k-1 there is an even number n in this sequence whose symmetric representation of sigma(n) has 2k-1 regions.

			a(5) = 153 = 17*3^2 is in the sequence and in the 5-column of the table since 1 < 2 < 3 < 6 < 3^2 < 17 = row(153) < 2*3^2 representing the 6 odd divisors 1 - 153 - 3 - 51 - 9 - 17 (see A237048) results in the following pattern for the widths of its 17 legs (see A249223): 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 5 regions with a single unit of width 2.
a(6) = 190 = 2*5*19 is in the sequence and in the 3-column of the table since 1 < 4 < 5 < 19 = row(190) < 4*5 representing the 4 odd divisors 1 - 190 - 5 - 19 results in the following pattern for the widths of its 19 legs: 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with a single unit of width 2.
Number 66 = 2*3*11 is not in the sequence since positions 1 < 3 < 4 < 11 = row(66) < 4*3 representing the 4 odd divisors 1 - 3 - 33 - 11 violate the condition 4 = 4*d_1 < d_2 = 3; its symmetric representation of sigma consists of a single region in which the third leg and its symmetric copy have width 2 in addition to a single unit of width 2 at the diagonal.

Cf. A000384, A000396, A116565, A129521, A237048, A237270, A237591, A237593, A249223, A338488.

(* function path[] and support functions are defined in A237270 *)
a338488[m_, n_] := Module[{p0=path[m-1], p1, k, srs, w2, list={}}, For[k=m, k<=n, k++, p1=path[k]; srs=Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[p1, 1], -1], p0}]]; w2=Length[Select[srs, #=={2, 2}&]]; If[Max[srs]==2&&w2==1, AppendTo[list, k]]; p0=p1]; list]
a338488[1,25000] (* sequence data *)

A226754 Numbers of the form pq, p and q prime with q=2p+3.

Original entry on oeis.org

14, 65, 119, 377, 629, 779, 1769, 3827, 4559, 5777, 9179, 10877, 16109, 19109, 25877, 32639, 37949, 39059, 49769, 56279, 60377, 75077, 78209, 79799, 100127, 103739, 105569, 145529, 154289, 161027, 189419, 228149, 244649, 250277, 288419, 294527, 316409, 335789

Offset: 1

José María Grau Ribas, Jun 16 2013

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A129521, A156592, A226755.

fa = FactorInteger; t[n_]:=Length[fa[n]] == 2 && fa[n][[1,2]]== fa[n][[2, 2]] == 1 && 2 fa[n][[1, 1]]+3 == fa[n][[2, 1]];Select[1+Range[200000], t]
Times@@#&/@Select[Table[{p,2p+3},{p,Prime[Range[200]]}],PrimeQ[#[[2]]]&] (* Harvey P. Dale, Jul 03 2021 *)

list(lim)=my(v=List(),q); forprime(p=2,(sqrt(8*lim+9)-3)\4, if(isprime(q=2*p+3),listput(v,p*q))); Vec(v) \\ Charles R Greathouse IV, Nov 19 2013

A252044 Numbers n such that s + 1/p = 0, where {d(i), i=1..q} are the q distinct prime divisors of n, s = Sum_{i=1..q} (-1)^(i+1)*i/d(i) and p = Product_{i=1..q} d(i).

Original entry on oeis.org

6, 12, 15, 18, 24, 36, 45, 48, 54, 72, 75, 91, 96, 108, 114, 135, 144, 162, 192, 216, 225, 228, 288, 324, 342, 375, 384, 405, 432, 456, 486, 576, 637, 648, 675, 684, 703, 768, 864, 912, 972, 1026, 1125, 1152, 1183, 1215, 1296, 1368, 1458, 1536, 1728, 1824, 1875

Offset: 1

Michel Lagneau, Dec 13 2014

nonn

The semiprimes p*q, p and q prime with q=2*p-1 (A129521) are in the sequence.

			18 is in the sequence because the prime factors of 18 are {2,3} => s = 1/2 - 2/3, 1/p = 1/6 and 1/2 - 2/3 + 1/6 = -1/6 + 1/6 = 0.
114 is in the sequence because the prime factors of 114 are {2,3,19} => s = 1/2 - 2/3 + 3/19, 1/p = 1/114 and 1/2 - 2/3 + 3/19 + 1/114 = -1/114 + 1/114 = 0.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A129521, A007947 (product of the distinct prime factors of n).

with(numtheory):nn:=10000:
for n from 1 to nn do:
   x:=factorset(n):n0:=nops(x):
   s:=sum('i*((-1)^(i+1))/x[i]','i'=1..n0):s0:=product('x[i]','i'=1..n0):
   p:=product('x[i]','i'=1..n0):s2:=s+1/s0:
    if s2=0
    then
    printf(`%d, `,n):
    else
    fi:
  od:

fQ[n_] := Block[{pd = First@# & /@ FactorInteger@ n, rng}, rng = Range@ Length@ pd; 1 == (Times @@ pd)*Total[rng/pd*((-1)^rng)]]; Select[ Range@ 2000, fQ@# &] (* Robert G. Wilson v, Jan 11 2015 *)

isok(n) = {my(vp = factor(n)[,1]~); 1/prod(i=1, #vp, vp[i]) + sum(i=1, #vp, (-1)^(i+1)*i/vp[i]) == 0;} \\ Michel Marcus, Jan 12 2015

A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

Original entry on oeis.org

15, 35, 45, 70, 77, 91, 110, 130, 135, 143, 154, 170, 182, 187, 190, 209, 221, 225, 238, 247, 266, 286, 299, 322, 323, 350, 374, 391, 405, 418, 437, 442, 493, 494, 506, 527, 550, 551, 572, 589, 598, 638, 646, 650, 667, 682, 703, 713, 748, 754, 782, 806, 814, 836, 850

Offset: 1

Hartmut F. W. Hoft, Oct 30 2020

nonn

This sequence is a subsequence of A279102. The definition of the sequence excludes squares of primes, A001248, since the 3 regions of their symmetric representation of sigma have width 1 (first column in the irregular triangle of A247687).
Table of numbers in this sequence arranged by the number of prime factors, counting multiplicities:
     2     3     4      5      6      7  ...
  ------------------------------------------
    15    45   135    405   1215   3645
    35    70   225   1125   5625    ...
    77   110   350   1750   8750 744795
    91   130   550   2584    ...    ...
   143   154   572   2750  85455
   187   170   650   3128    ...
   209   182   748   3250
   221   190   836   3496
   247   238   850   3944
   299   266   884   4216
   ...   ...   ...    ...
              1035   9585
               ...    ...
The numbers in the first row of the table above are b(k) = 5*3^k, k>=1, (see A005030) so that infinitely many odd numbers occur outside of the first column. The central region of the symmetric representation of sigma(b(k)) contains 2*k-1 separate contiguous sections consisting of sequences of entire legs of width 2, k>=1 (see Lemma 2 in the link).
Conjecture: The combined extent of these sections in sigma(b(k)) is 2*3^(k-1) - 1 = A048473(k-1), k>=1.
Since each number n in the first column and first row has a prime factor of odd exponent a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 2. For odd numbers n not in the first row or column in which all prime factors have even powers, such as 225 and 5625 in the second row, a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 1 (see Lemma 1 in the link).
For each k>=3 and every prime p such that b(k-1) < 2*p < 4*b(k-2), the odd number p*b(k-1) is in the column of b(k). The two inequalities are equivalent to b(k-1) <= row(p*b(k-1)) < 2*b(k-1) ensuring that the symmetric representation of sigma(p*b(k-1)) consists of 3 regions.
45 is the only odd number in its column (see Lemma 3 in the link).
Since the factors of n = p*q satisfy 2 < p < q < 2*p the first column in the table above is a subsequence of A082663 and of A087718 (see Lemma 4 in the link). Each of the two outer regions consists of a single leg of width 1 and length (1 + p*q)/2. The center region of size p+q consists of two subparts (see A196020 & A280851) of width 1 of sizes 2*p-q and 2*q-p, respectively (see Lemma 5 in the link). The table below arranges the first column in the table above according to the length 2*p-q of their single contiguous extent of width 2 in the center region:
     1     3      5      7      9     11     13     15  ...
------------------------------------------------------
    15    35    187    247    143    391   2257    323
    91    77    493    589    221   1363   3139    437
   703   209    943   2479    551   2911   6649    713
  1891   299   1537   3397    851   3901    ...   1247
  2701   527   4183   8509   1643   6313          1457
  ...    ...    ...    ...    ...    ...          ....
A129521: p*q satisfies 2*p - q = 1 (excluding A129521(1)=6)
A226755: p*q satisfies 2*p - q = 3 (excluding A226755(1)=9)
Sequences with larger differences 2*p - q are not in OEIS.

			a(6) = 91 = 7*13 is in the sequence and in the 2-column of the first table since 1 < 2 < 7 < 13 = row(91) representing the 4 odd divisors 1 - 91 - 7 - 13 (see A237048) results in the following pattern for the widths of the legs (see A249223): 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2. It also is in the 1-column of the second table since it has a single area of width 2 which is 1 unit long.
a(29) = 405 = 5*3^4 is in the sequence and in the 5-column of the first table since 1 < 2 < 3 < 5 < 6 < 9 < 10 < 15 < 18 < 27 = row(405) representing the 10 odd divisors 1 - 405 - 3 - 5 - 135 - 9 - 81 - 15 - 45 - 27 results in the following pattern for the widths of the legs: 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2, and 7 = 2*4 - 1 sections of width 2 in the central region.
a(35) = 506 = 2*11*23 is in the sequence since positions 1 < 4 < 11 < 23 < row(506) = 31 representing the 4 odd divisors 1 - 253 - 11 - 23 results in the following pattern for the widths of the legs: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2 for 3 regions with width not exceeding 2, with the two outer regions consisting of 3 legs of width 1, and a single area of width 2 in the central region.

Hartmut F. W. Hoft, proofs for quoted lemmas

Cf. A001248, A005030, A048473, A082663, A087718, A129521, A196020, A226755, A235791, A237048, A237270, A237271, A237591, A237593, A247687, A249223, A279102, A280107, A280851.

(* Functions path and a237270 are defined in A237270 *)
maxDiagonalLength[n_] := Max[Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]]]
a338486[m_, n_] := Module[{r, list={}, k}, For[k=m, k<=n, k++, r=a237270[k]; If[Length[r]== 3 && maxDiagonalLength[k]==2,AppendTo[list, k]]]; list]
a338486[1, 850]

A349497 a(n) is the smallest element in the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Nov 20 2021

nonn

			a(2) = 1 since the continued fraction of the harmonic mean of the divisors of 2, 4/3 = 1 + 1/3, has 2 elements, {1, 3}, and the smallest of them is 1.

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

Cf. A001248, A001599, A001600, A099377, A099378, A129521, A349473.

a[n_] := Min[ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]]; Array[a, 100]

a(p) = 1 for a prime p.
a(p^2) = 1 for a prime p != 3.
a(A129521(n)) = 1 for n > 3.
For a harmonic number m = A001599(k), a(m) = A099377(m) = A001600(k).

A380007 Hexagonal numbers that are sphenic numbers.

Original entry on oeis.org

66, 190, 231, 435, 561, 861, 946, 1653, 2278, 3655, 4371, 5151, 5995, 6441, 8911, 9453, 10011, 10585, 13366, 15051, 15753, 16471, 20301, 21115, 22366, 22791, 23653, 26335, 32131, 33153, 39621, 40186, 45451, 50403, 54946, 62481, 69751, 72771, 77421, 80601, 83845, 93961, 99235, 102831

Offset: 1

Massimo Kofler, Jan 08 2025

nonn

			66 = 2*3*11 is the product of 3 distinct primes and the 6th hexagonal number hex(6) = 6*(2*6-1).
231 = 3*7*11 is the product of 3 distinct primes and the 11th hexagonal number hex(11) = 11*(2*11-1).

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

Intersection of A000384 and A007304.

Cf. A129521.

semiQ[k_] := FactorInteger[k][[;; , 2]] == {1, 1}; q[k_] := (PrimeQ[k] && semiQ[2*k - 1]) || (PrimeQ[2*k - 1] && semiQ[k]); Table[k*(2*k - 1), {k, Select[Range[250], q]}] (* Amiram Eldar, Jan 08 2025 *)

A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.

Original entry on oeis.org

1, 6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 60, 66, 72, 75, 84, 90, 91, 96, 108, 120, 132, 135, 144, 150, 162, 168, 180, 192, 198, 216, 225, 240, 252, 264, 270, 276, 288, 300, 306, 312, 324, 330, 336, 360, 375, 384, 396, 405, 420, 432, 435, 450, 480, 486, 504

Offset: 1

William Rex Marshall, Oct 24 2013

nonn

Equivalently, the numbers with exactly one divisor that is not in an arithmetic progression of at least three divisors.

Contains p^j*(2*p-1)^k for j,k>=1 if p and 2*p-1 are primes. - Robert Israel, Apr 13 2020

			36 appears in this sequence because its proper divisors are 1, 2, 3, 4, 6, 9, 12 and 18, each of which appears in at least one of the following arithmetic progressions of at least three proper divisors of 36: {1, 2, 3, 4}, {3, 6, 9, 12}, {6, 12, 18}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A091010, A091011.

Contains A033845, A129521.

filter:= proc(n) local S,D,tau,a,b;
  S:= numtheory:-divisors(n) minus {n};
  D:= sort(convert(S,list));
  tau:= nops(D);
  for a from 1 to tau-2 do for b from a+1 to tau-1 do
    if member(2*D[b]-D[a],D) then
      S:= S minus {D[a],D[b],2*D[b]-D[a]};
      if S = {} then return true fi;
    fi
  od od;
  false;
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Apr 13 2020

filterQ[n_] := Module[{S, D, tau, a, b}, S = Most @ Divisors[n]; D = S; tau = Length[D]; For[a = 1, a <= tau - 2, a++, For[b = a + 1, b <= tau - 1, b++, If [MemberQ[D, 2 D[[b]] - D[[a]]], S = S ~Complement~ {D[[a]], D[[b]], 2 D[[b]] - D[[a]]}; If[S == {}, Return[True]]]]]; False];
filterQ[1] = True;
Select[Range[1000], filterQ] (* Jean-François Alcover, Sep 26 2020, after Robert Israel *)

cp's OEIS Frontend

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