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A379592 Number of coreful divisor pairs (d, k/d), d | k, d < k/d, such that only one divisor divides the other, where k is in A320966.

%I A379592 #6 Jan 01 2025 01:47:43
%S A379592 1,1,1,2,2,1,1,1,1,3,1,1,1,2,3,2,1,1,1,1,2,1,4,2,1,2,1,2,1,2,2,1,2,1,
%T A379592 4,1,3,3,1,1,1,1,2,2,3,1,1,2,2,1,5,3,1,3,1,1,1,4,1,1,1,1,2,1,2,2,3,1,
%U A379592 1,3,1,1,2,4,1,2,5,1,1,1,4,1,1,2,5,1,1
%N A379592 Number of coreful divisor pairs (d, k/d), d | k, d < k/d, such that only one divisor divides the other, where k is in A320966.
%C A379592 Number of ways to write k = A320966(n) as a product of numbers i and j, i < j, such that rad(i) = rad(j) = rad(k), and either i | j or j | i, where rad = A007947 is the squarefree kernel.
%C A379592 Analogous to A370329, where the reference domain is A001694 instead of A320966.
%H A379592 Michael De Vlieger, <a href="/A379592/b379592.txt">Table of n, a(n) for n = 1..10000</a>
%e A379592 Let s(n) = A320966(n).
%e A379592 a(1) = 1 since s(1) = 8 = 2*4.
%e A379592 a(2) = 1 since s(2) = 16 = 2*8.
%e A379592 a(3) = 1 since s(3) = 27 = 3*9.
%e A379592 a(4) = 2 since s(4) = 32 = 2*16 = 4*8.
%e A379592 a(10) = 3 since s(10) = 128 = 2*64 = 4*32 = 8*16.
%e A379592 a(23) = 4 since s(23) = 512 = 2*256 = 4*128 = 8*64 = 16*32.
%e A379592 a(181) = 7 since s(181) = 20736 = 6*3456 = 12*1728 = 18*1152 = 24*864 = 36*576 = 48*432 = 72*288, etc.
%t A379592 nn = 5400; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A379592 s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
%t A379592   Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] > 0 &];
%t A379592 Table[k = s[[n]];
%t A379592   Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
%t A379592     _?(And[rad[#1] == rad[#2],
%t A379592        Xor[Divisible[#2, #1],
%t A379592            Divisible[#1, #2]]] & @@ # &)], {n, Length[s]}]
%Y A379592 Cf. A320966, A370329, A379552.
%K A379592 nonn
%O A379592 1,4
%A A379592 _Michael De Vlieger_, Dec 28 2024