cp's OEIS Frontend

A379772 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d} and either rad(d) | k/d or rad(k/d) | d, where k = A378767(n).

%I A379772 #5 Jan 09 2025 19:16:29
%S A379772 1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,2,2,1,1,1,1,1,1,
%T A379772 1,1,2,1,1,1,1,2,1,1,1,1,1,2,2,1,1,1,1,3,1,1,1,1,2,1,2,1,2,1,1,1,1,2,
%U A379772 2,1,1,1,2,1,1,1,1,2,2,1,1,1,1,2,1,1,1
%N A379772 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d} and either rad(d) | k/d or rad(k/d) | d, where k = A378767(n).
%C A379772 Let rad = A007947 and let omega = A001221.
%C A379772 Number of ways to write k = A378767(n) as a product of numbers i and j, omega(i) < omega(j) = omega(i*j), that are neither coprime nor divide one another, where rad(i) | j, but rad(j) does not divide i. Both i and j are necessarily composite.
%H A379772 Michael De Vlieger, <a href="/A379772/b379772.txt">Table of n, a(n) for n = 1..10000</a>
%e A379772 Let s(n) = A378767(n).
%e A379772 a(1) = 1 since s(1) = 24 = 4*6, omega(4) < omega(6) = omega(24), rad(4) | 6.
%e A379772 a(2) = 1 since s(2) = 40 = 4*10, omega(4) < omega(10) = omega(40), rad(4) | 10.
%e A379772 a(3) = 1 since s(3) = 48 = 6*8, omega(8) < omega(6) = omega(48), rad(8) | 6.
%e A379772 a(9) = 2 since s(9) = 96 = 6*16 = 8*12.
%e A379772 a(54) = 3 since s(54) = 384 = 6*64 = 12*32 = 16*24.
%e A379772 a(165) = 5 since s(165) = 1080 = 4*270 = 9*120 = 12*90 = 18*60 = 30*36.
%t A379772 nn = 120; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A379772 s = Select[Select[Range[nn],
%t A379772   AnyTrue[FactorInteger[#][[All, -1]], # > 2 &] &],
%t A379772     Not @* PrimePowerQ];
%t A379772 Table[k = s[[n]];
%t A379772   Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
%t A379772     _?( (m = GCD @@ {##};
%t A379772       And[! MemberQ[{1, #1, #2}, m],
%t A379772         And[PrimeNu[#1] < PrimeNu[#2],
%t A379772           Divisible[#2, rad[#1]]] & @@
%t A379772           SortBy[{##}, PrimeNu]]) & @@ # &)], {n, Length[s]}]
%Y A379772 Cf. A001221, A007947, A378767.
%K A379772 nonn
%O A379772 1,9
%A A379772 _Michael De Vlieger_, Jan 02 2025