A378900 -id:A378900 - cp's OEIS frontend

A378769 Intersection of A375055 and A376936.

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

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

cp's OEIS Frontend

A378769 Intersection of A375055 and A376936.

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