A379552 - OEIS

%I #9 Dec 28 2024 09:14:20

%S 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,

%T 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,

%U 1,2,1,4,2,2,1,2,3,1,4,2,1,1,2,1,1,3,4

%N 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.

%C 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.

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

%H Michael De Vlieger, <a href="/A379552/b379552.txt">Table of n, a(n) for n = 1..10000</a>

%e 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:

%e a(1) = 1 since b(1) = 216 = 12*18.

%e a(2) = 1 since b(2) = 432 = 18*24.

%e a(3) = 1 since b(3) = 648 = 12*54.

%e a(4) = 2 since b(4) = 864 = 18*48 = 24*36.

%e a(14) = 3 since b(14) = 3456 = 18*192 = 36*96 = 48*72.

%e a(22) = 4 since b(22) = 7776 = 24*324 = 48*162 = 54*144 = 72*108, etc.

%t nn = 2^16;

%t rad[x_] := Times @@ FactorInteger[x][[All, 1]];

%t s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],

%t   Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];

%t Table[k = s[[n]];

%t   Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],

%t     _?(And[1 < GCD @@ {##},

%t        rad[#1] == rad[#2],

%t        Mod[#1, #2] != 0,

%t        Mod[#2, #1] != 0] & @@ # &)], {n, Length[s]}]

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

%K nonn,easy,new

%O 1,4

%A _Michael De Vlieger_, Dec 25 2024