A379592 - OEIS

%I #6 Jan 01 2025 01:47:43

%S 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 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 1,3,1,1,2,4,1,2,5,1,1,1,4,1,1,2,5,1,1

%N 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 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 Analogous to A370329, where the reference domain is A001694 instead of A320966.

%H Michael De Vlieger, <a href="/A379592/b379592.txt">Table of n, a(n) for n = 1..10000</a>

%e Let s(n) = A320966(n).

%e a(1) = 1 since s(1) = 8 = 2*4.

%e a(2) = 1 since s(2) = 16 = 2*8.

%e a(3) = 1 since s(3) = 27 = 3*9.

%e a(4) = 2 since s(4) = 32 = 2*16 = 4*8.

%e a(10) = 3 since s(10) = 128 = 2*64 = 4*32 = 8*16.

%e a(23) = 4 since s(23) = 512 = 2*256 = 4*128 = 8*64 = 16*32.

%e a(181) = 7 since s(181) = 20736 = 6*3456 = 12*1728 = 18*1152 = 24*864 = 36*576 = 48*432 = 72*288, etc.

%t nn = 5400; 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 &] > 0 &];

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

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

%t     _?(And[rad[#1] == rad[#2],

%t        Xor[Divisible[#2, #1],

%t            Divisible[#1, #2]]] & @@ # &)], {n, Length[s]}]

%Y Cf. A320966, A370329, A379552.

%K nonn,new

%O 1,4

%A _Michael De Vlieger_, Dec 28 2024