A379772 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d} and either rad(d) | k/d or rad(k/d) | d, where k = A378767(n).

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

Offset: 1

Author

Michael De Vlieger, Jan 02 2025

Keywords

• nonn

Comments

Let rad = A007947 and let omega = A001221.

Number of ways to write k = A378767(n) as a product of numbers i and j, omega(i) < omega(j) = omega(i*j), that are neither coprime nor divide one another, where rad(i) | j, but rad(j) does not divide i. Both i and j are necessarily composite.

Examples

			Let s(n) = A378767(n).
a(1) = 1 since s(1) = 24 = 4*6, omega(4) < omega(6) = omega(24), rad(4) | 6.
a(2) = 1 since s(2) = 40 = 4*10, omega(4) < omega(10) = omega(40), rad(4) | 10.
a(3) = 1 since s(3) = 48 = 6*8, omega(8) < omega(6) = omega(48), rad(8) | 6.
a(9) = 2 since s(9) = 96 = 6*16 = 8*12.
a(54) = 3 since s(54) = 384 = 6*64 = 12*32 = 16*24.
a(165) = 5 since s(165) = 1080 = 4*270 = 9*120 = 12*90 = 18*60 = 30*36.
		

Links

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

Crossrefs

Cf. A001221, A007947, A378767.

Programs

Mathematica

nn = 120; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Select[Select[Range[nn],
  AnyTrue[FactorInteger[#][[All, -1]], # > 2 &] &],
    Not @* PrimePowerQ];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
    _?( (m = GCD @@ {##};
      And[! MemberQ[{1, #1, #2}, m],
        And[PrimeNu[#1] < PrimeNu[#2],
          Divisible[#2, rad[#1]]] & @@
          SortBy[{##}, PrimeNu]]) & @@ # &)], {n, Length[s]}]