cp's OEIS Frontend

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.

%I A379552 #9 Dec 28 2024 09:14:20
%S A379552 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 A379552 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 A379552 1,2,1,4,2,2,1,2,3,1,4,2,1,1,2,1,1,3,4
%N 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.
%C A379552 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 A379552 Divisors d and k/d are both composite, further, are neither squarefree nor prime powers, hence in A126706.
%H A379552 Michael De Vlieger, <a href="/A379552/b379552.txt">Table of n, a(n) for n = 1..10000</a>
%e A379552 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 A379552 a(1) = 1 since b(1) = 216 = 12*18.
%e A379552 a(2) = 1 since b(2) = 432 = 18*24.
%e A379552 a(3) = 1 since b(3) = 648 = 12*54.
%e A379552 a(4) = 2 since b(4) = 864 = 18*48 = 24*36.
%e A379552 a(14) = 3 since b(14) = 3456 = 18*192 = 36*96 = 48*72.
%e A379552 a(22) = 4 since b(22) = 7776 = 24*324 = 48*162 = 54*144 = 72*108, etc.
%t A379552 nn = 2^16;
%t A379552 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A379552 s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
%t A379552   Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
%t A379552 Table[k = s[[n]];
%t A379552   Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
%t A379552     _?(And[1 < GCD @@ {##},
%t A379552        rad[#1] == rad[#2],
%t A379552        Mod[#1, #2] != 0,
%t A379552        Mod[#2, #1] != 0] & @@ # &)], {n, Length[s]}]
%Y A379552 Cf. A007947, A126706, A376936, A379553, A379554.
%K A379552 nonn,easy
%O A379552 1,4
%A A379552 _Michael De Vlieger_, Dec 25 2024