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A380032 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and d | k/d but rad(k/d) does not divide d, where k is in A126706.

%I A380032 #6 Jan 15 2025 08:43:12
%S A380032 1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,2,1,2,2,1,1,
%T A380032 1,1,1,1,1,1,1,3,1,1,1,1,1,1,2,2,1,1,1,1,1,2,3,1,1,1,3,2,1,2,1,1,2,1,
%U A380032 2,1,2,2,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1
%N A380032 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) > 1 and d | k/d but rad(k/d) does not divide d, where k is in A126706.
%C A380032 Number of ways to write k = A126706(n) as a product of noncoprime numbers i and j, i < j, where i | j, but rad(j) does not divide i. This is to say that j has a factor that does not divide i, hence omega(i) < omega(j) = omega(k).
%C A380032 Divisor d may be prime, but k/d is composite.
%H A380032 Michael De Vlieger, <a href="/A380032/b380032.txt">Table of n, a(n) for n = 1..10000</a>
%e A380032 Let s(n) = A126706(n).
%e A380032 a(1) = 1 since s(1) = 12 = 2*6.
%e A380032 a(2) = 1 since s(2) = 18 = 3*6.
%e A380032 a(3) = 1 since s(3) = 20 = 2*10.
%e A380032 a(4) = 1 since s(4) = 24 = 2*12.
%e A380032 a(6) = 2 since s(6) = 36 = 2*18 = 3*12.
%e A380032 a(42) = 3 since s(42) = 144 = 2*72 = 3*48 = 4*36.
%e A380032 a(204) = 4 since s(204) = 576 = 2*288 = 3*192 = 4*144 = 8*72.
%e A380032 a(257) = 5 since s(257) = 720 = 2*360 = 3*240 = 4*180 = 6*120 = 12*60, etc.
%t A380032 nn = 120;
%t A380032 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A380032 s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
%t A380032 Table[k = s[[n]];
%t A380032   Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
%t A380032     _?( (m = GCD @@ {##};
%t A380032       And[! MemberQ[{1, #2}, m],
%t A380032       m == #1,
%t A380032       ! Divisible[#1, rad[#2]] ] ) & @@ # &)], {n, Length[s]}]
%Y A380032 Cf. A001221, A025487, A126706.
%K A380032 nonn
%O A380032 1,6
%A A380032 _Michael De Vlieger_, Jan 11 2025