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A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.

%I A379465 #41 Jan 19 2025 09:29:32
%S A379465 30,42,66,126,70,78,198,264,90,234,126,120,462,312,270,270,306,150,
%T A379465 154,696,798,936,390,210,210,290,210,714,210,286,210,462,744,240,1710,
%U A379465 1224,910,330,420,2262,390,270,714,870,1050,294,330,630,630,2232,378,1620,330
%N A379465 Sum of coreful divisors d | k such that gcd(d, k/d) is not in {1, d, k/d} and rad(d) = rad(k/d), where k is in A376936 and rad = A007947.
%C A379465 Define function f(x) to be 1/2 * card({ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) }), a counting function of distinct divisor pairs (d, x/d) that are both coreful but neither divides the other.
%C A379465 Define function g(x) to be Sum_{ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) } d.
%C A379465 Define h(x) to be equal to A364988(x) = Sum_{ d | x : rad(d) = rad(x) } d, sum of coreful divisors of x.
%C A379465 a(n) = g(A376936(n)) <= h(A376936(n)).
%C A379465 The function f(x) is analogous to tau(x) = A000005(x) while g(x) is analogous to sigma(x) = A000203(x).
%C A379465 f(k) > 0 and g(k) > 1 for k in A376936, otherwise f(k) = 0 and g(k) = 0.
%C A379465 Since rad(d) = rad(k/d) = rad(k), a(n) = m*rad(k), with integer m > 1.
%H A379465 Michael De Vlieger, <a href="/A379465/b379465.txt">Table of n, a(n) for n = 1..10000</a>
%e A379465 Let s = A376936.
%e A379465 a(1) = 30 since s(1) = 216 = 12*18 = 2*6 + 3*6 = 5*rad(216), and the sum of these is 30.
%e A379465 a(2) = 42 since s(2) = 432 = 18*24 = 3*6 + 4*6 = 7*rad(432), and the sum of these is 42.
%e A379465 a(3) = 66 since s(3) = 648 = 12*54 = 2*6 + 9*6 = 11*rad(648), and their sum is 66.
%e A379465 a(4) = 126 since s(4) = 864 = 18*48 = 24*36, and the sum of all these divisors is 126, etc. Note that 18 + 48 + 24 + 36 = 3*6 + 8*6 + 4*6 + 6*6 = 21*rad(864).
%t A379465 nn = 2^16;
%t A379465 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A379465 s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
%t A379465   Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
%t A379465 Map[Function[n,
%t A379465   DivisorSum[n, # &,
%t A379465     And[! MemberQ[{1, #1, #2}, GCD @@ {##}],
%t A379465       rad[#1] == rad[#2]] & @@ {#, n/#} &]], s]
%Y A379465 Cf. A364988, A379552.
%K A379465 nonn
%O A379465 1,1
%A A379465 _Michael De Vlieger_, Jan 13 2025