cp's OEIS Frontend

A379752 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and neither rad(d) | k/d nor rad(k/d) | d, where k is in A375055.

%I A379752 #7 Jan 03 2025 13:41:57
%S A379752 1,1,1,2,1,1,1,1,1,2,2,1,1,1,1,1,3,2,1,2,2,1,2,1,2,1,1,2,1,3,1,1,1,1,
%T A379752 3,1,1,2,1,2,2,1,3,2,1,2,2,1,2,1,4,1,1,1,3,1,2,1,1,3,1,3,1,2,1,3,1,1,
%U A379752 1,1,2,2,3,2,2,1,3,3,1,1,1,3,1,4,2,2,1
%N A379752 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and neither rad(d) | k/d nor rad(k/d) | d, where k is in A375055.
%C A379752 Number of ways to write k = A375055(n) as a product of numbers i and j, i < j, that are neither coprime nor divide one another, where each has a factor that does not divide the other. Such numbers i and j are necessarily composite.
%H A379752 Michael De Vlieger, <a href="/A379752/b379752.txt">Table of n, a(n) for n = 1..10000</a>
%e A379752 Let s(n) = A375055(n).
%e A379752 a(1) = 1 since s(1) = 60 = 6 * 10 = (2*3) * (2*5).
%e A379752 a(2) = 1 since s(2) = 84 = 6 * 14 = (2*3) * (2*7).
%e A379752 a(3) = 1 since s(3) = 90 = 6 * 15 = (2*3) * (3*5).
%e A379752 a(4) = 2 since s(4) = 120 = 6*20 = 10*12.
%e A379752 a(17) = 3 since s(17) = 240 = 6*40 = 10*24 = 12*20.
%e A379752 a(51) = 4 since s(51) = 480 = 6*80 = 10*48 = 12*40 = 20*24.
%e A379752 a(117) = 5 since s(117) = 840 = 6*140 = 10*84 = 12*70 = 14*60 = 20*42 = 28*30, etc.
%t A379752 nn = 120;
%t A379752 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A379752 s = Select[Range[nn], PrimeOmega[#] > PrimeNu[#] > 2 & ];
%t A379752 Table[k = s[[n]];
%t A379752   Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
%t A379752     _?(And[1 < GCD @@ {##},
%t A379752        Nor[Divisible[#2, rad[#1]],
%t A379752            Divisible[#1, rad[#2]] ] ] & @@ # &)], {n, Length[s]}]
%Y A379752 Cf. A375055.
%K A379752 nonn,easy
%O A379752 1,4
%A A379752 _Michael De Vlieger_, Jan 01 2025