A376936 -id:A376936 - cp's OEIS frontend

A379552 Number of pairs (d, k/d), d < k/d, such that d|k, rad(d) = rad(k/d) = rad(k), but d|k/d, for k = A376936(n), where rad = A007947.

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 2, 1, 2, 1, 1, 2, 3, 4, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 4, 1, 2, 1, 3, 4, 1, 2, 6, 1, 3, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 4, 2, 2, 1, 2, 3, 1, 4, 2, 1, 1, 2, 1, 1, 3, 4

Offset: 1

Michael De Vlieger, Dec 25 2024

In other words, one half the number of coreful complementary divisor pairs (d, k/d), d|k, that do not divide one another, for k in A376936, the sequence of numbers k that have at least 1 such pair.

Divisors d and k/d are both composite, further, are neither squarefree nor prime powers, hence in A126706.

			Let b(n) = A376936(n) and define property Q pertaining to (d, k/d), d|k, to be rad(d) = rad(k/d) = rad(k) but neither d | k/d nor k/d | d. Examples below show only (d, k/d) that have property Q:
a(1) = 1 since b(1) = 216 = 12*18.
a(2) = 1 since b(2) = 432 = 18*24.
a(3) = 1 since b(3) = 648 = 12*54.
a(4) = 2 since b(4) = 864 = 18*48 = 24*36.
a(14) = 3 since b(14) = 3456 = 18*192 = 36*96 = 48*72.
a(22) = 4 since b(22) = 7776 = 24*324 = 48*162 = 54*144 = 72*108, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A126706, A376936, A379553, A379554.

nn = 2^16;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       rad[#1] == rad[#2],
       Mod[#1, #2] != 0,
       Mod[#2, #1] != 0] & @@ # &)], {n, Length[s]}]

A378769 Intersection of A375055 and A376936.

Original entry on oeis.org

5400, 9000, 10584, 10800, 13500, 16200, 18000, 21168, 21600, 24696, 26136, 27000, 31752, 32400, 36000, 36504, 37044, 40500, 42336, 43200, 45000, 48600, 49000, 49392, 52272, 54000, 62424, 63504, 64800, 67500, 68600, 72000, 73008, 74088, 77976, 78408, 81000, 84672

Offset: 1

Michael De Vlieger, Dec 13 2024

Let omega = A001221, bigomega = A001222, rad = A007947.
Powerful numbers k with bigomega(k) > omega(k) > 2 that are divisible by two distinct prime cubes p^3 and q^3.
Numbers k such that there exists (d, k/d), d | k, such that d neither divides nor is coprime to k/d and vice versa in the following 3 ways:
Type A: rad(d) does not divide d/k and rad(d/k) does not divide d
Type B: rad(d) divides d/k but rad(d/k) does not divide d
Type C: rad(d) | d/k and rad(d/k) | d, hence rad(d) = rad(d/k) = rad(k), a kind of coreful divisor pair.
Since (d, d/k) are noncoprime and do not divide one another, both must be composite, thus k is also composite.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.
A378767 = { k : omega(k) > 1, p^3 | k for some prime p }, and
A376936 = { k : rad(k)^2 | k, p^3 | k and q^3 | k for distinct primes p, q }.
Therefore, we need only take intersection of A375055 and A376936.

			Table of the first 12 terms of this sequence, showing examples of types A, B, and C described in Comments.
   n     a(n)  Factors of a(n)    Type A      Type B      Type C
  ----------------------------------------------------------------
   1    5400   2^3 * 3^3 * 5^2    24 * 225    4 * 1350    60 * 90
   2    9000   2^3 * 3^2 * 5^3    18 * 500    4 * 2250    60 * 150
   3   10584   2^3 * 3^3 * 7^2    24 * 441    4 * 2646    84 * 126
   4   10800   2^4 * 3^3 * 5^2    48 * 225    8 * 1350    90 * 120
   5   13500   2^2 * 3^3 * 5^3    12 * 1125   9 * 1500    90 * 150
   6   16200   2^3 * 3^4 * 5^2    24 * 675    4 * 4050    60 * 270
   7   18000   2^4 * 3^2 * 5^3    18 * 1000   8 * 2250   120 * 150
   8   21168   2^4 * 3^3 * 7^2    48 * 441    8 * 2646   126 * 168
   9   21600   2^5 * 3^3 * 5^2    50 * 432    8 * 2700    90 * 240
  10   24696   2^3 * 3^2 * 7^3    18 * 1372   4 * 6174    84 * 294
  11   26136   2^3 * 3^3 * 11^2   24 * 1089   4 * 6534   132 * 198
  12   27000   2^3 * 3^3 * 5^3    24 * 1125   4 * 6750    60 * 450

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..16, showing divisor pairs of type A in light gray, type B in blue and gold, and type C in black.

Cf. A001694, A024619, A126706, A375055, A376936, A378767, A378900.

Cf. A082020, A082695.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^16],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[s, PrimeOmega[#] > PrimeNu[#] > 2 &]

Intersection of A375055, A376936, and A378767.
This sequence is { k : rad(k)^2 | k, bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) - ((Sum_{p prime} (1/(p^2*(p-1))))^2 - Sum_{p prime} (1/(p^4*(p-1)^2)))/2 = 0.0025524144364532126894... . - Amiram Eldar, Dec 21 2024

A379553 Numbers k in A376936 that set records in A379552.

Original entry on oeis.org

216, 864, 3456, 7776, 31104, 124416, 279936, 497664, 972000, 1944000, 3888000, 7776000, 11664000, 15552000, 31104000, 34992000, 46656000, 62208000, 77760000, 97200000, 194400000, 291600000, 388800000, 777600000, 874800000, 1166400000, 1555200000, 3110400000, 3499200000

Offset: 1

Michael De Vlieger, Dec 25 2024

nonn

Proper subset of A025487.

			Let b(n) = A376936(n) and define property Q pertaining to (d, k/d), d|k, to be rad(d) = rad(k/d) = rad(k) but neither d | k/d nor k/d | d. Table below shows prime power decomposition of a(n), n = 1..12, writing only exponents in the "exp." column:
   n       a(n)   exp.   b(n)  (d,a(n)/d) with property Q
  -----------------------------------------------------------------
   1       216    3.3      1   (12,18)
   2       864    5.3      2   (18,48), (24,36)
   3      3456    7.3      3   (18,192), (36,96), (48,72)
   4      7776    5.5      4   (24,324), (48,162), (54,144), (72,108)
   5     31104    7.5      6
   6    124416    9.5      8
   7    279936    7.7      9
   8    497664   11.5     10
   9    972000    5.5.3   12
  10   1944000    6.5.3   14
  11   3888000    7.5.3   18
  12   7776000    8.5.3   20
See expanded table in links.

Michael De Vlieger, Table of n, a(n) for n = 1..212 (a(n) <= A002110(19).)
Michael De Vlieger, Prime power decomposition of a(n), n = 1..212.

Cf. A376936, A379552, A379554.

r = 0; nn = 10^9;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]; nn = Length[s];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
    rad[#1] == rad[#2],
    Mod[#1, #2] != 0,
    Mod[#2, #1] != 0] & @@ # &)], {i, nn}] ][[-1, 1]]

A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.

Original entry on oeis.org

30, 42, 66, 126, 70, 78, 198, 264, 90, 234, 126, 120, 462, 312, 270, 270, 306, 150, 154, 696, 798, 936, 390, 210, 210, 290, 210, 714, 210, 286, 210, 462, 744, 240, 1710, 1224, 910, 330, 420, 2262, 390, 270, 714, 870, 1050, 294, 330, 630, 630, 2232, 378, 1620, 330

Offset: 1

Michael De Vlieger, Jan 13 2025

nonn

Define function f(x) to be 1/2 * card({ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) }), a counting function of distinct divisor pairs (d, x/d) that are both coreful but neither divides the other.
Define function g(x) to be Sum_{ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) } d.
Define h(x) to be equal to A364988(x) = Sum_{ d | x : rad(d) = rad(x) } d, sum of coreful divisors of x.
a(n) = g(A376936(n)) <= h(A376936(n)).
The function f(x) is analogous to tau(x) = A000005(x) while g(x) is analogous to sigma(x) = A000203(x).
f(k) > 0 and g(k) > 1 for k in A376936, otherwise f(k) = 0 and g(k) = 0.
Since rad(d) = rad(k/d) = rad(k), a(n) = m*rad(k), with integer m > 1.

			Let s = A376936.
a(1) = 30 since s(1) = 216 = 12*18 = 2*6 + 3*6 = 5*rad(216), and the sum of these is 30.
a(2) = 42 since s(2) = 432 = 18*24 = 3*6 + 4*6 = 7*rad(432), and the sum of these is 42.
a(3) = 66 since s(3) = 648 = 12*54 = 2*6 + 9*6 = 11*rad(648), and their sum is 66.
a(4) = 126 since s(4) = 864 = 18*48 = 24*36, and the sum of all these divisors is 126, etc. Note that 18 + 48 + 24 + 36 = 3*6 + 8*6 + 4*6 + 6*6 = 21*rad(864).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A364988, A379552.

nn = 2^16;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Map[Function[n,
  DivisorSum[n, # &,
    And[! MemberQ[{1, #1, #2}, GCD @@ {##}],
      rad[#1] == rad[#2]] & @@ {#, n/#} &]], s]

A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

Original entry on oeis.org

24, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272, 276, 280, 288, 294

Offset: 1

Michael De Vlieger, Dec 24 2024

Both divisors d and d/k are composite, since primes p either divide or are coprime to another number, and all numbers smaller than p are coprime to p.
Proper subset of A126706; contains A378769, which in turn contains A378984.
Consider composite k, m, k != m. Define a "neutral" relation to be such that 1 < gcd(k,m) and not equal to either k or m. Then neither k nor m divides the other, and k and m are not coprime. If k is neutral to m, then m is neutral to k, since order does not matter. Then either the squarefree kernel of one divides the other or it does not. Thus, there are 3 kinds of neutral relation:
Type A: Though gcd(k,m) > 1, k has a factor P that does not divide m, and m has a factor Q that does not divide k.
Type B: rad(k) = rad(m), yet neither k divides m nor m divides k, where rad = A007947 is the squarefree kernel.
Type C: Squarefree kernel of one number divides the other, while the other has a factor that does not divide the former.
A378769, subset of this sequence, contains numbers k that have all 3 types of neutral relation between at least 1 divisor pair (d, k/d) for each.

			a(1) = 24 = 2^3 * 3 = 4*6, both composite; gcd(4,6) = 2, 4 does not divide 6 (type C).
a(2) = 40 = 2^3 * 5 = 4*10, gcd(4,10) = 2 (type C).
a(3) = 48 = 2^4 * 3 = 6*8, gcd(6,8) = 2 (type C).
a(6) = 60 = 2^2 * 3 * 5 = 6*10, gcd(6,10) = 2 (type A).
a(12) = 96 = 2^5 * 3 = 6*16 = 8*12, both type C.
a(38) = 216 = 2^3 * 3^3 = 4*54 (type C) = 9*24 (type C) = 12*18 (type B)
a(1605) = 5400 = 2^3 * 3^3 * 5^2 = 4*1350 (type C) = 24*225 (type A) = 60*90 (type B) = A378769(1).
a(10475) = 32400 = 2^4 * 3^4 * 5^2 = 8*4050 (type C) = 48*675 (type A) = 120*270 (type B) = A378984(1) = A378769(14), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Chart of (d, a(n)/d) for a(n) = 1..144, showing only the smallest d for each type of neutral relation, where type A is shown in gray, type B in black, and type C in either blue or gold.

Cf. A045763, A085986, A126706, A375055, A376271, A376936, A378767, A378769, A378984.

nn = 300; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
Complement[
  Select[Range[nn], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
  Union[Reap[
    While[p <= mm, q = NextPrime[p];
      While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];
        p = NextPrime[p] ] ][[-1, 1]] ]^2 ]

This sequence is A376271 \ A085986 = {k : bigomega(k) > omega(k) > 1, bigomega(k) > 3} \ { k^2 : bigomega(k) = omega(k) = 2 }, where bigomega = A001222 and omega = A001221.

Union of A375055, A376936, and A378767.

A379554 Records in A379552.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 18, 20, 21, 24, 26, 27, 28, 30, 32, 36, 40, 42, 48, 52, 54, 56, 60, 64, 72, 78, 80, 90, 96, 100, 108, 114, 120, 126, 128, 135, 136, 144, 150, 160, 168, 170, 176, 180, 186, 192, 200, 204, 210, 224, 240, 248, 252, 264, 272, 280, 288

Offset: 1

Michael De Vlieger, Dec 25 2024

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..212

Cf. A376936, A379552, A379553.

r = 0; nn = 10^9;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]; nn = Length[s];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[r]] &@
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
    rad[#1] == rad[#2],
    Mod[#1, #2] != 0,
    Mod[#2, #1] != 0] & @@ # &)], {i, nn}] ][[-1, 1]]

A378900 Squares of numbers divisible by the squares of two distinct primes.

Original entry on oeis.org

1296, 5184, 10000, 11664, 20736, 32400, 38416, 40000, 46656, 50625, 63504, 82944, 90000, 104976, 129600, 153664, 156816, 160000, 186624, 194481, 202500, 219024, 234256, 250000, 254016, 291600, 331776, 345744, 360000, 374544, 419904, 455625, 456976, 467856, 490000

Offset: 1

Michael De Vlieger, Dec 12 2024

Also, the squares in A376936.
Proper subset of A378767, in turn a proper subset of A286708, the intersection of A001694 and A024619.
Numbers that have 3 kinds of coreful divisor pairs (d, k/d), d | k, i.e., rad(d) = rad(k/d) = rad(k) where rad = A007947. These kinds are described as follows:
Type A: d = k/d, which pertain to square k (in A000290).
Type B: d | k/d, d < k/d, which pertain to k in A320966, powerful numbers divisible by a cube.
Type C: neither d | k/d nor k/d | d, which pertain to k in A376936.
Since divisors d, k/d may either divide or not divide the other, there are no other cases.
In addition the following kinds of divisor pairs are also seen:
Type D: (d, k/d) such that d | k/d but there exists a factor Q | k/d that does not divide d. Then omega(d) < omega(k/d) = omega(k).
Type E: Nontrivial unitary divisor pairs (d, k/d) such that gcd(d, k/d) = 1, d > 1, k/d > 1. Let prime power factor p^m | k be such that m is maximized. Then set d = p^m and it is clear that for any k in A024619, there exists at least 1 nontrivial unitary divisor pair.

			Let b = A036785.
Table of the first 12 terms of this sequence, showing examples of types A, B, and C of coreful pairs of divisors.
   n    a(n)   Factors of a(n)    b(n)   Type B       Type C
  -------------------------------------------------------------
   1    1296   2^4  * 3^4          36    6 * 216      24 * 54
   2    5184   2^6  * 3^4          72    6 * 864      48 * 108
   3   10000   2^4  * 5^4         100   10 * 1000     40 * 250
   4   11664   2^4  * 3^6         108    6 * 1944     24 * 486
   5   20736   2^8  * 3^4         144    6 * 3456     54 * 384
   6   32400   2^4  * 3^4 * 5^2   180   30 * 1080    120 * 270
   7   38416   2^4  * 7^4         196   14 * 2744     56 * 686
   8   40000   2^6  * 5^4         200   10 * 4000     80 * 500
   9   46656   2^6  * 3^6         216    6 * 7776     48 * 972
  10   50625   3^4  * 5^4         225   15 * 3375    135 * 375
  11   63504   2^4  * 3^4 * 7^2   252   42 * 1512    168 * 378
  12   82944   2^10 * 3^4         288    6 * 13824    54 * 1536

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Listing of select divisor pairs of a(n), n = 1..12, showing divisor pairs of type A in light gray, type B in orange and purple, and type C in black.

Cf. A000290, A001694, A007947, A024619, A036785, A126706, A286708, A320966, A376936, A378768.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}],   IntegerQ@ Sqrt[#] &] &[500000];
Union@ Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

a(n) = A036785(n)^2.

Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} 1/(p^4-1)) = 0.0015294876575980711757... . - Amiram Eldar, Dec 21 2024

A378984 Squares in A378769.

Original entry on oeis.org

32400, 63504, 90000, 129600, 156816, 202500, 219024, 254016, 291600, 345744, 360000, 374544, 467856, 490000, 518400, 571536, 627264, 685584, 777924, 810000, 876096, 960400, 1016064, 1089936, 1166400, 1210000, 1245456, 1382976, 1411344, 1440000, 1498176, 1587600

Offset: 1

Michael De Vlieger, Dec 15 2024

Let omega = A001221, bigomega = A001222, and rad = A007947.
Numbers k that have all types of divisor pairs (d, k/d), d | k, that are listed in both A378769 and A378900. These are listed below:
Type A*: (Nontrivial) unitary divisor pairs, i.e., d coprime to k/d. The rest of the types are in cototient.
Type B*: gcd(d, k/d) > 1, rad(d) !| k/d, rad(k/d) !| d. These exist for k in A375055.
Type C: d < k/d, d | k/d but rad(k/d) !| d. Implies rad(k/d) = rad(k) and omega(d) < omega(k/d). These exist for k in A126706.
Type D: Either rad(d) | k/d, rad(k/d) !| d or vice versa. These exist for k in A378767.
Type E*: d = k/d = sqrt(k).
Type F: rad(d) = rad(k/d) = rad(k), d < k/d, d | k/d. These exist for k in A320966.
Type G*: rad(d) = rad(k/d) = rad(k), neither d | k/d nor k/d | d. These exist for k in A376936.
Asterisks denote symmetric types.
Since numbers d and k/d are either coprime or not, and if not, the squarefree kernel of one either divides the other or not, and if so, d divides k/d or not, and if so, d = k/d or not, there are no other types.
Smallest odd term is a(45) = 2480625.
Square roots not A350372: sqrt(810000) = 900 is not in A350372.

			a(1) = 32400 = 2^4 * 3^4 * 5^2 has the following divisor pair types:
  Type A: 16 * 2025, Type B: 48 * 675, Type C: 2 * 16200, Type D: 8 * 4050
  Type E: 180 * 180, Type F: 30 * 1080, Type G: 120 * 270.
a(2) = 63504 = 2^4 * 3^4 * 7^2 has the following divisor pair types:
  Type A: 16 * 3969, Type B: 48 * 1323, Type C: 2 * 31752, Type D: 8 * 7938
  Type E: 252 * 252, Type F: 42 * 1512, Type G: 168 * 378.
a(3) = 90000 = 2^4 * 3^2 * 5^4 has the following divisor pair types:
  Type A: 9 * 10000, Type B: 18 * 5000, Type C: 2 * 45000, Type D: 8 * 11250
  Type E: 300 * 300, Type F: 30 * 3000, Type G: 120 * 750, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram listing all divisor pairs for a(n), n = 1..8, showing type A in white, type B in light gray, type C in green or red, type D in blue or gold, type E in dark gray, type F in orange or purple, and type G in black.
Michael De Vlieger, Diagram listing divisor pairs (d, k/d) for k = a(n), n = 1..60, showing only those with the smallest d and using the same color scheme as above, for each type and its reversal if the type is nonsymmetric.

Cf. A000290, A001221, A001222, A007947, A126706, A320966, A350372, A375055, A376936, A378767, A378769, A378900.

Cf. A013661, A082020.

s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[2^21],  IntegerQ@ Sqrt[#] &];
t = Select[s, Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Select[t, PrimeOmega[#] > PrimeNu[#] > 2 &]

This sequence is { k = s^2 : rad(k)^2 | k,
  bigomega(k) > omega(k) > 2, p^3 | k and q^3 | k for distinct primes p, q }.
Intersection of A378769 and A378900.
Intersection of A000290, A375055, and A376936.
Sum_{n>=1} = Pi^2/6 - (15/Pi^2) * (1 + Sum_{p prime} (1/(p^4-1))) - ((Sum_{p prime} (1/(p^2*(p^2-1))))^2 - Sum_{p prime} (1/(p^4*(p^2-1)^2)))/2 = 0.00015490158528995570146... . - Amiram Eldar, Dec 21 2024

A376687 Numbers that set records in in A376281.

Original entry on oeis.org

24, 96, 120, 240, 360, 480, 840, 1080, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 27720, 30240, 55440, 60480, 83160, 110880, 151200, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 831600, 997920, 1330560, 1441440, 1663200, 2162160, 2882880

Offset: 1

Michael De Vlieger, Jan 08 2025

nonn

Proper subset of the intersection A025487 and A379336.
There are three kinds of pairs (d, k/d), d | k, such that gcd(d, k/d) does not equal 1, d, or k/d:
Type A: rad(d) does not divide k/d, and rad(k/d) does not divide d (see A379752), where rad = A007947.
Type B: the squarefree kernel of one divisor divides the other but the reverse is not true (see A379772).
Type C: rad(d) = rad(k/d), i.e., d, k/d, and k are coreful (see A379552).
Conjecture: Numbers k that set records in A376281 do not have type C divisor pairs, i.e., those that are coreful but neither divides the other. This, since type C requires k to be powerful and divisible by cubes of 2 distinct primes (i.e., in A376936). Therefore the record is achieved only through large numbers of type A and B.
Since type A divisor pairs are common for composite k in A375055, this sequence is resembles A379752.
Since d and k/d are both composite, this sequence resembles A059992.
This sequence, to a lesser extent A379752, and a greater extent A059992, contains many highly composite numbers. (See plot of S(n) = union of this sequence and A002182 below, and corresponding graphs in respective other sequences.)

			Let b(n) = A376281(n).
Table showing exponents of prime power factors of a(n) for n = 1..20.
Example: a(5) = 360 = 2^3 * 3^2 * 5, hence we write "3.2.1".
   n    a(n)  Exp.   b(a(n))
  ----------------------------------
   1     24 *   3.1        1   4*6
   2     96     5.1        2   6*16 = 8*12
   3    120 **  3.1.1      3   4*30 = 6*20 = 10*12
   4    240 *   4.1.1      4   6*40 = 8*30 = 10*24 = 12*20
   5    360 **  3.2.1      5   4*90 = 10*36 = 12*30 = 15*24 = 18*20
   6    480     5.1.1      6   6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24
   7    840 *   3.1.1.1    7
   8   1080     3.3.1      9
   9   1680 *   4.1.1.1   10
  10   2160     4.3.1     11
  11   2520 **  3.2.1.1   13
  12   3360     5.1.1.1   14
*  = a(n) is highly composite (in A002182),
** = a(n) is superior highly composite (in both A002182 and A002201).

Michael De Vlieger, Table of n, a(n) for n = 1..241
Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.
Michael De Vlieger, List of (d, k/d), d < k/d, k = a(n), n = 1..24, such that gcd(d, k/d) is not in {1, d, k/d}, showing type A in light gray, type B in either blue or gold, and type C (if it occurs) in black.
Michael De Vlieger, Prime power decomposition of a(n), n = 1..241.

Cf. A002182, A002201, A007947, A025487, A059992, A379336, A376281, A377525, A379752.

(* Load function f at A025487 *)
r = 0; s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not@*SquareFreeQ];
nn = Length[s]; Print[nn]
Reap[Monitor[
  Do[k = s[[i]];
    If[# > r, r = #; Sow[k]] &@
      Count[Transpose@{#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
        _?(And[1 < GCD @@ {##}, Mod[#1, #2] != 0,
           Mod[#2, #1] != 0] & @@ # &)], {i, nn}], i] ][[-1, 1]]

A377846 Powerful numbers that are not divisible by the cubes of more than one distinct prime.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 441, 484, 500, 512, 529, 576, 625, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Amiram Eldar, Nov 09 2024

Subsequence of A377821 and first differs from it at n = 33: A377821(33) = 432 = 2^4 * 3^3 is not a term of this sequence.

Numbers whose prime factorization has exponents that are all larger than 1 and no more than one exponent is larger than 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Complement of A376936 within A001694.
Subsequence of A377821.
Subsequences: A143610, A377847.
Cf. A082020.

q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, Length[e] == 1 || e[[-2]] == 2]; With[{max = 1300}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # == 1 || q[#] &]]

is(k) = if(k == 1, 1, my(e = vecsort(factor(k)[, 2])); e[1] > 1 && (#e == 1 || e[#e - 1] == 2));

Sum_{n>=1} 1/a(n) = (15/Pi^2) * (1 + Sum_{p prime} 1/((p-1)*(p^2+1))) = 1.92240214785252516795... .

cp's OEIS Frontend

A379552 Number of pairs (d, k/d), d < k/d, such that d|k, rad(d) = rad(k/d) = rad(k), but d|k/d, for k = A376936(n), where rad = A007947.

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A378769 Intersection of A375055 and A376936.

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A379553 Numbers k in A376936 that set records in A379552.

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A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.

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A379336 Numbers k such that there exists a divisor pair (d, d/k) such that one neither divides nor is coprime to the other.

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A379554 Records in A379552.

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A378900 Squares of numbers divisible by the squares of two distinct primes.

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A378984 Squares in A378769.

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