A380032 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and d | k/d but rad(k/d) does not divide d, where k is in A126706.

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

Offset: 1

Author

Michael De Vlieger, Jan 11 2025

Keywords

• nonn

Comments

Number of ways to write k = A126706(n) as a product of noncoprime numbers i and j, i < j, where i | j, but rad(j) does not divide i. This is to say that j has a factor that does not divide i, hence omega(i) < omega(j) = omega(k).

Divisor d may be prime, but k/d is composite.

Examples

			Let s(n) = A126706(n).
a(1) = 1 since s(1) = 12 = 2*6.
a(2) = 1 since s(2) = 18 = 3*6.
a(3) = 1 since s(3) = 20 = 2*10.
a(4) = 1 since s(4) = 24 = 2*12.
a(6) = 2 since s(6) = 36 = 2*18 = 3*12.
a(42) = 3 since s(42) = 144 = 2*72 = 3*48 = 4*36.
a(204) = 4 since s(204) = 576 = 2*288 = 3*192 = 4*144 = 8*72.
a(257) = 5 since s(257) = 720 = 2*360 = 3*240 = 4*180 = 6*120 = 12*60, etc.
		

Links

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

Crossrefs

Cf. A001221, A025487, A126706.

Programs

Mathematica

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