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

1,1

Comments

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.

Links

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.

Formula

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

Example

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

Mathematica

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 &]

Crossrefs

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

Cf. A082020, A082695.

Sequence in context: A328562 A340109 A364991 * A190106 A252572 A382451

Adjacent sequences: A378766 A378767 A378768 * A378770 A378771 A378772

Keyword

nonn,easy

Author

Michael De Vlieger, Dec 13 2024

Status

approved