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A380143 Sum of divisors d | k such that d and k/d share factors but both have a factor that does not divide the other, where k is in A375055.

%I A380143 #23 Jan 19 2025 09:29:41
%S A380143 16,20,21,48,27,28,24,25,32,60,55,39,40,32,44,45,112,65,36,84,84,52,
%T A380143 72,35,91,57,36,96,36,140,44,63,64,45,123,40,68,108,48,85,120,75,172,
%U A380143 96,80,136,132,56,95,48,240,49,88,48,141,92,108,93,50,196,52,172
%N A380143 Sum of divisors d | k such that d and k/d share factors but both have a factor that does not divide the other, where k is in A375055.
%C A380143 In other words, sum of divisors d | k such that gcd(d, k/d) > 1 but neither rad(d) | k/d nor rad(k/d) | d, where rad = A007947 and k is in A375055.
%C A380143 Define quality Q pertaining to 2 natural numbers a and b such that gcd(a, b) > 1 but neither rad(a) | b nor rad(b) | a.
%C A380143 Define function f(x) = A379752 to be the cardinality of divisor pairs (d, x/d) that have quality Q. f(x) > 0 for x in A375055, otherwise f(x) = 0.
%H A380143 Michael De Vlieger, <a href="/A380143/b380143.txt">Table of n, a(n) for n = 1..10000</a>
%H A380143 Michael De Vlieger, <a href="/A380143/a380143.png">Log log scatterplot of a(n)</a>, n = 1..2^16.
%e A380143 Let s = A375055.
%e A380143 a(1) = 16 since s(1) = 60 = 6*10; 6 + 10 = 16.
%e A380143 a(2) = 20 since s(2) = 84 = 6*14; 6 + 14 = 20.
%e A380143 a(3) = 21 since s(3) = 90 = 6*15; 6 + 15 = 21.
%e A380143 a(4) = 48 since s(4) = 120 = 6*20 = 10*12; 6 + 20 + 10 + 12 = 48, etc.
%t A380143 nn = 540; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A380143 s = Select[Range[nn], PrimeOmega[#] > PrimeNu[#] > 2 & ];
%t A380143 Table[k = s[[n]];
%t A380143   DivisorSum[k, # &,
%t A380143     And[1 < GCD @@ {##},
%t A380143       Nor[Divisible[#2, rad[#1] ],
%t A380143           Divisible[#1, rad[#2] ] ] ] & @@
%t A380143     {#, k/#} &], {n, Length[s]}]
%Y A380143 Cf. A007947, A375055, A379752.
%K A380143 nonn
%O A380143 1,1
%A A380143 _Michael De Vlieger_, Jan 18 2025