A379552 Number of pairs (d, k/d), d < k/d, such that d|k, rad(d) = rad(k/d) = rad(k), but d|k/d, for k = A376936(n), where rad = A007947.

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 2, 1, 2, 1, 1, 2, 3, 4, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 4, 1, 2, 1, 3, 4, 1, 2, 6, 1, 3, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 4, 2, 2, 1, 2, 3, 1, 4, 2, 1, 1, 2, 1, 1, 3, 4

Offset: 1

Author

Michael De Vlieger, Dec 25 2024

Keywords

• nonn
• easy

Comments

In other words, one half the number of coreful complementary divisor pairs (d, k/d), d|k, that do not divide one another, for k in A376936, the sequence of numbers k that have at least 1 such pair.

Divisors d and k/d are both composite, further, are neither squarefree nor prime powers, hence in A126706.

Examples

			Let b(n) = A376936(n) and define property Q pertaining to (d, k/d), d|k, to be rad(d) = rad(k/d) = rad(k) but neither d | k/d nor k/d | d. Examples below show only (d, k/d) that have property Q:
a(1) = 1 since b(1) = 216 = 12*18.
a(2) = 1 since b(2) = 432 = 18*24.
a(3) = 1 since b(3) = 648 = 12*54.
a(4) = 2 since b(4) = 864 = 18*48 = 24*36.
a(14) = 3 since b(14) = 3456 = 18*192 = 36*96 = 48*72.
a(22) = 4 since b(22) = 7776 = 24*324 = 48*162 = 54*144 = 72*108, etc.
		

Links

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

Crossrefs

Cf. A007947, A126706, A376936, A379553, A379554.

Programs

Mathematica

nn = 2^16;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Table[k = s[[n]];
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       rad[#1] == rad[#2],
       Mod[#1, #2] != 0,
       Mod[#2, #1] != 0] & @@ # &)], {n, Length[s]}]