A375055 -id:A375055 - cp's OEIS frontend

A379752 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and neither rad(d) | k/d nor rad(k/d) | d, where k is in A375055.

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

Offset: 1

Michael De Vlieger, Jan 01 2025

Number of ways to write k = A375055(n) as a product of numbers i and j, i < j, that are neither coprime nor divide one another, where each has a factor that does not divide the other. Such numbers i and j are necessarily composite.

			Let s(n) = A375055(n).
a(1) = 1 since s(1) = 60 = 6 * 10 = (2*3) * (2*5).
a(2) = 1 since s(2) = 84 = 6 * 14 = (2*3) * (2*7).
a(3) = 1 since s(3) = 90 = 6 * 15 = (2*3) * (3*5).
a(4) = 2 since s(4) = 120 = 6*20 = 10*12.
a(17) = 3 since s(17) = 240 = 6*40 = 10*24 = 12*20.
a(51) = 4 since s(51) = 480 = 6*80 = 10*48 = 12*40 = 20*24.
a(117) = 5 since s(117) = 840 = 6*140 = 10*84 = 12*70 = 14*60 = 20*42 = 28*30, etc.

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

Cf. A375055.

nn = 120;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[nn], PrimeOmega[#] > PrimeNu[#] > 2 & ];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       Nor[Divisible[#2, rad[#1]],
           Divisible[#1, rad[#2]] ] ] & @@ # &)], {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

A380143 Sum of divisors d | k such that d and k/d share factors but both have a factor that does not divide the other, where k is in A375055.

Original entry on oeis.org

16, 20, 21, 48, 27, 28, 24, 25, 32, 60, 55, 39, 40, 32, 44, 45, 112, 65, 36, 84, 84, 52, 72, 35, 91, 57, 36, 96, 36, 140, 44, 63, 64, 45, 123, 40, 68, 108, 48, 85, 120, 75, 172, 96, 80, 136, 132, 56, 95, 48, 240, 49, 88, 48, 141, 92, 108, 93, 50, 196, 52, 172

Offset: 1

Michael De Vlieger, Jan 18 2025

nonn

In other words, sum of divisors d | k such that gcd(d, k/d) > 1 but neither rad(d) | k/d nor rad(k/d) | d, where rad = A007947 and k is in A375055.
Define quality Q pertaining to 2 natural numbers a and b such that gcd(a, b) > 1 but neither rad(a) | b nor rad(b) | a.
Define function f(x) = A379752 to be the cardinality of divisor pairs (d, x/d) that have quality Q. f(x) > 0 for x in A375055, otherwise f(x) = 0.

			Let s = A375055.
a(1) = 16 since s(1) = 60 = 6*10; 6 + 10 = 16.
a(2) = 20 since s(2) = 84 = 6*14; 6 + 14 = 20.
a(3) = 21 since s(3) = 90 = 6*15; 6 + 15 = 21.
a(4) = 48 since s(4) = 120 = 6*20 = 10*12; 6 + 20 + 10 + 12 = 48, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16.

Cf. A007947, A375055, A379752.

nn = 540; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[nn], PrimeOmega[#] > PrimeNu[#] > 2 & ];
Table[k = s[[n]];
  DivisorSum[k, # &,
    And[1 < GCD @@ {##},
      Nor[Divisible[#2, rad[#1] ],
          Divisible[#1, rad[#2] ] ] ] & @@
    {#, k/#} &], {n, Length[s]}]

A178212 Nonsquarefree numbers divisible by exactly three distinct primes.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset: 1

Reinhard Zumkeller, May 24 2010

nonn

			60 is in the sequence because it is not squarefree and it is divisible by three distinct primes: 2, 3, 5.
72 is not in the sequence, because although it is not squarefree, it is divisible by only two distinct primes: 2 and 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

A subsequence of A033987.
A085987 is a subsequence.
Cf. A007304, A200511, A200521.
Cf. A000005, A001221, A001222, A013929, A085987, A123712.
Cf. A027748, A124010.
Subsequence of A375055, which differs starting at a(43) = 440 > A375055(43) = 420.

a178212 n = a178212_list !! (n-1)
a178212_list = filter f [1..] where
   f x = length (a027748_row x) == 3 && any (> 1) (a124010_row x)
-- Reinhard Zumkeller, Apr 03 2015

nsD3Q[n_] := Block[{fi = FactorInteger@ n}, Length@ fi == 3 && Plus @@ Last /@ fi > 3]; Select[ Range@ 494, nsD3Q] (* Robert G. Wilson v, Feb 09 2012 *)
Select[Range[500], PrimeNu[#] == 3 && PrimeOmega[#] > 3 &] (* Alonso del Arte, Mar 23 2015, based on a comment from Robert G. Wilson v, Feb 09 2012; requires Mathematica 7.0+ *)

is_A178212(n)={ omega(n)==3 & bigomega(n)>3 }
for(n=1,999,is_A178212(n) & print1(n",")) \\ M. F. Hasler, Feb 09 2012

A001221(a(n)) = 3; A001222(a(n)) > 3; A000005(n) >= 12;

a(n) = A123712(n) for n <= 52, possibly more.

A379097 Numbers that are not waterproof.

Original entry on oeis.org

60, 84, 120, 132, 156, 168, 204, 228, 240, 264, 276, 280, 300, 312, 315, 336, 348, 372, 408, 420, 440, 444, 456, 480, 492, 495, 516, 520, 528, 552, 560, 564, 585, 588, 600, 616, 624, 630, 636, 660, 672, 680, 693, 696, 708, 728, 732, 744, 760, 765, 780, 804, 816

Offset: 1

Peter Luschny, Dec 16 2024

nonn

Zero and one are waterproof numbers by convention. Numbers that admit a prime factorization are not waterproof if their water capacity is > 0. (The water capacity of a number is defined in A275339.)

Proper subset of A375055, in turn a proper subset of A126706, since A001221(a(n)) >= 3 and a maximum multiplicity is required for at least one prime power factor, so as to have positive water capacity. - Michael De Vlieger, Dec 18 2024

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

Cf. A275339, A379094, A379096, A379098.

# The function 'water_capacity' is defined in A275339.
is_not_waterproof := n -> ifelse(n < 2, false, is(water_capacity(n) <> 0)):
select(is_not_waterproof, [seq(0..820)]);

nn = 816;
s = Select[Range[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
Select[s, Function[f, And[NoneTrue[{Sort[f], ReverseSort[f]}, # == f &],
  Total[(f //. {a___, b_, c__, d_, e___} /;
    AllTrue[{c}, And[# < b, # < d] &] :>
    {a, b, Sequence @@ Table[Min[b, d], {Length[{c}]}], d, e}) - f] > 0] ]
[Power @@@ FactorInteger[#]] &] (* Michael De Vlieger, Dec 18 2024, after Jean-François Alcover at A275339 *)

# The function 'WaterCapacity' is defined in A275339.
print([n for n in range(818) if WaterCapacity(n) > 0])

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.

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]]

A379753 Numbers that set records in A379752.

Original entry on oeis.org

60, 120, 240, 480, 840, 1260, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 27720, 36960, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 443520, 498960, 554400, 665280, 720720, 997920, 1081080, 1330560, 1441440, 2162160, 2882880, 3603600, 4324320, 5765760

Offset: 1

Michael De Vlieger, Jan 01 2025

nonn

Proper subset of the intersection of A025487 and A375055.
Conjecture: subset of A332785 = A126706 \ A286708.
This sequence seems to be rich in highly composite numbers, the prime shape of a(n) resembles that of highly composite numbers, with long tails of large prime factors with multiplicity 1.
Terms not in A002182 are not all of the form 2^5 * prime(i..j), 1 < i < j, for example, a(24) = 443520 = 2^7 * 3^2 * 5 * 7 * 11.

			Let b(n) = A379752(n).
Table showing exponents of prime power factors of a(n) for n = 1..12.
Example: a(6) = 1260 = 2^2 * 3^2 * 5 * 7, hence we write "2.2.1.1".
   n      a(n)       Exp.    b(a(n))
  ----------------------------------
   1       60 **   2.1.1        1   6*10
   2      120 **   3.1.1        2   6*20 = 10*12
   3      240 *    4.1.1        3   6*40 = 10*24 = 12*20
   4      480      5.1.1        4   6*80 = 10*48 = 12*40 = 20*24
   5      840 *    3.1.1.1      6   6*140 = 10*84 = 12*70 = 14*60 = 20*42 = 28*30
   6     1260 *    2.2.1.1      7
   7     1680 *    4.1.1.1      9
   8     2520 **   3.2.1.1     11
   9     3360      5.1.1.1     12
  10     5040 **   4.2.1.1     15
  11     7560 *    3.3.1.1     16
  12    10080 *    5.2.1.1     19
*  = 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..226
Michael De Vlieger, Prime Power Decomposition of a(n) for n = 1..226, expanding on the table in the example.
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.

Cf. A002182, A002201, A025487, A375055, A379752, A379754.

(* Load function f at A025487 *)
r = 0;
s = Select[Union@ Flatten@ f[14][[4 ;; -1]], Not@*SquareFreeQ];
nn = Length[s]; Print[nn];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       Nor[Divisible[#2, rad[#1]],
           Divisible[#1, rad[#2]] ] ] & @@ # &)], {i, nn}] ][[-1, 1]]

A380432 Numbers k such that bigomega(k) > omega(k) > 3.

Original entry on oeis.org

420, 630, 660, 780, 840, 924, 990, 1020, 1050, 1092, 1140, 1170, 1260, 1320, 1380, 1386, 1428, 1470, 1530, 1540, 1560, 1596, 1638, 1650, 1680, 1710, 1716, 1740, 1820, 1848, 1860, 1890, 1932, 1950, 1980, 2040, 2070, 2100, 2142, 2184, 2220, 2244, 2280, 2340, 2380

Offset: 1

Michael De Vlieger, Jan 29 2025

A200521 is a proper subset; a(144) = 4620 is not contained in A200521; A200521(144) = 4650.

Subset of A375055, which is in turn a subset of A126706.

			Table of select a(n) showing exponents listed in columns of primes written vertically in the heading. For p^0 we instead write "." for clarity:
              Exponent of prime
                     1 1 1 1
   n   a(n)   2 3 5 7 1 3 7 9
  -----------------------------
    1    420   2 1 1 1
    2    630   1 2 1 1
    3    660   2 1 1 . 1
    4    780   2 1 1 . . 1
    5    840   3 1 1 1
    6    924   2 1 . 1 1
    7    990   1 2 1 . 1
    8   1020   2 1 1 . . . 1
    9   1050   1 1 2 1
   10   1092   2 1 . 1 . 1
   11   1140   2 1 1 . . . . 1
   12   1170   1 2 1 . . 1
  144   4620   2 1 1 1 1
  190   5460   2 1 1 1 . 1
  275   6930   1 2 1 1 1

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

Cf. A001221, A001222, A126706, A200521, A375055.

Select[Range[2500], PrimeOmega[#] > PrimeNu[#] > 3 &]

cp's OEIS Frontend

A379752 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and neither rad(d) | k/d nor rad(k/d) | d, where k is in A375055.

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A378769 Intersection of A375055 and A376936.

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A380143 Sum of divisors d | k such that d and k/d share factors but both have a factor that does not divide the other, where k is in A375055.

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A178212 Nonsquarefree numbers divisible by exactly three distinct primes.

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A379097 Numbers that are not waterproof.

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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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A378984 Squares in A378769.

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A376687 Numbers that set records in in A376281.

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