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.

30, 42, 66, 126, 70, 78, 198, 264, 90, 234, 126, 120, 462, 312, 270, 270, 306, 150, 154, 696, 798, 936, 390, 210, 210, 290, 210, 714, 210, 286, 210, 462, 744, 240, 1710, 1224, 910, 330, 420, 2262, 390, 270, 714, 870, 1050, 294, 330, 630, 630, 2232, 378, 1620, 330

Offset: 1

Author

Michael De Vlieger, Jan 13 2025

Keywords

• nonn

Comments

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.

Define function g(x) to be Sum_{ d | x : gcd(d, x/d) not in {1, d, x/d}, rad(d) = rad(x) } d.

Define h(x) to be equal to A364988(x) = Sum_{ d | x : rad(d) = rad(x) } d, sum of coreful divisors of x.

a(n) = g(A376936(n)) <= h(A376936(n)).

The function f(x) is analogous to tau(x) = A000005(x) while g(x) is analogous to sigma(x) = A000203(x).

f(k) > 0 and g(k) > 1 for k in A376936, otherwise f(k) = 0 and g(k) = 0.

Since rad(d) = rad(k/d) = rad(k), a(n) = m*rad(k), with integer m > 1.

Examples

			Let s = A376936.
a(1) = 30 since s(1) = 216 = 12*18 = 2*6 + 3*6 = 5*rad(216), and the sum of these is 30.
a(2) = 42 since s(2) = 432 = 18*24 = 3*6 + 4*6 = 7*rad(432), and the sum of these is 42.
a(3) = 66 since s(3) = 648 = 12*54 = 2*6 + 9*6 = 11*rad(648), and their sum is 66.
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).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A364988, A379552.

Programs

Mathematica

nn = 2^16;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &];
Map[Function[n,
  DivisorSum[n, # &,
    And[! MemberQ[{1, #1, #2}, GCD @@ {##}],
      rad[#1] == rad[#2]] & @@ {#, n/#} &]], s]