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A376281 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d}, where k is in A379336.

%I A376281 #14 Jan 11 2025 03:47:19
%S A376281 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,1,1,1,1,1,2,1,3,1,2,1,1,2,1,
%T A376281 1,1,1,3,1,2,1,1,1,4,1,1,2,1,3,3,1,1,3,2,1,1,1,2,1,1,1,3,1,2,1,1,4,1,
%U A376281 1,1,1,1,1,2,5,1,1,1,1,1,3,1,3,1,2,1,1
%N A376281 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d}, where k is in A379336.
%C A376281 Number of ways to write k = A379336(n) as a product of numbers i and j that are neither coprime nor does one number divide the other. Both i and j are necessarily composite.
%C A376281 Both i and j = k/i appear in row k of A133995.
%H A376281 Michael De Vlieger, <a href="/A376281/b376281.txt">Table of n, a(n) for n = 1..10000</a>
%e A376281 Let s(n) = A379336(n).
%e A376281 a(1) = 1 since s(1) = 24 = 4*6.
%e A376281 a(2) = 1 since s(2) = 40 = 4*10.
%e A376281 a(3) = 1 since s(3) = 48 = 6*8.
%e A376281 a(12) = 2 since s(12) = 96 = 6*16 = 8*12.
%e A376281 a(16) = 3 since s(16) = 120 = 4*30 = 6*20 = 10*12.
%e A376281 a(44) = 4 since s(44) = 240 = 6*40 = 8*30 = 10*24 = 12*20.
%e A376281 a(75) = 5 since s(75) = 360 = 4*90 = 10*36 = 12*30 = 15*24 = 18*20.
%e A376281 a(105) = 6 since s(105) = 480 = 6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24, etc.
%t A376281 nn = 500; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
%t A376281 s = Complement[
%t A376281   Select[Range[nn],
%t A376281     And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
%t A376281   Union[Reap[
%t A376281     While[p <= mm, q = NextPrime[p];
%t A376281       While[p*q <= mm, If[p != q, Sow[p*q]]; q = NextPrime[q]];
%t A376281         p = NextPrime[p]] ][[-1, 1]] ]^2 ];
%t A376281 Table[k = s[[n]];
%t A376281   1/2*DivisorSum[k, 1 &, ! MemberQ[{1, #1, #2}, GCD[#1, #2]] & @@ {#, k/#} &],
%t A376281   {n, Length[s]}]
%Y A376281 Cf. A045763, A133995, A379336, A379552, A379752, A379772.
%K A376281 nonn
%O A376281 1,12
%A A376281 _Michael De Vlieger_, Jan 08 2025