A379523 Sum of coreful divisors d | k such that gcd(d, k/d) > 1 and rad(d) = rad(k/d), with d | k/d and d < d/k, where k is in A320966 and rad = A007947.

6, 10, 12, 30, 54, 18, 30, 24, 30, 126, 30, 30, 42, 120, 238, 90, 60, 56, 42, 50, 126, 60, 510, 162, 130, 168, 60, 336, 70, 150, 234, 66, 240, 110, 990, 90, 378, 432, 84, 132, 78, 112, 210, 270, 546, 90, 110, 456, 330, 150, 2046, 1092, 182, 714, 102, 350, 260

Offset: 1

Author

Michael De Vlieger, Jan 15 2025

Keywords

• nonn

Comments

Alternatively, sum of divisors d | k such that omega(d) = omega(k/d) = omega(k), d | k/d, and d < k/d, where omega = A001221.

Examples

			a(1) = 6 since s(1) = 8 = 2*4; 2 | 4 but 4 > 2; 2+4 = 6. a(1) = (2^3-1)/(2-1)-1 = 6.
a(2) = 10 since s(2) = 16 = 2*8; 2 | 8 but 8 > 2; 2+8 = 10. a(2) = (2^4-1)/(2-1)-(2^2)-1 = 10.
a(3) = 12 since s(3) = 27 = 3*9; 3 | 9 but 9 > 3; 3+9 = 12. a(3) = (3^3-1)/(3-1)-1 = 12.
a(4) = 30 since s(4) = 32 = (2^5-1)/(2-1)-1 = 30.
a(5) = 54 since s(5) = 64 = (2^6-1)/(2-1)-(2^3)-1 = 54.
a(6) = 18 since s(6) = 72 = 6*12; 6 | 12 but 12 > 6; 6+12 = 18.
a(8) = 24 since s(8) = 108 = 6*18; 6 | 24 but 24 > 6; 6+18 = 24, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n) n = 1..10^5.

Crossrefs

Cf. A000079, A001221, A007947, A246549, A320966, A364988.

Programs

Mathematica

nn = 2500;
s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] > 0 &];
Map[Function[n,
  DivisorSum[n, # &,
    And[PrimeNu[#1] == PrimeNu[#2] == #3,
        Xor[Divisible[#2, #1], Divisible[#1, #2]]] & @@
    {#, n/#, PrimeNu[n]} &]], s]

Formula

Let s(n) = A320966(n).

a(n) <= A364988(s(n)).

Using Iverson brackets:

For s(n) = 2^e, e > 2, (i.e., in A000079):

a(n) = 2^e - [e mod 2 = 0]*sqrt(2^e)-2

= 2^e - 2*(e-1) for even e or 2^e-2 for odd e.

= A364988(s(n)) - [e mod 2 = 0]*2^(e/2) for s(n) = 2^e, e > 2.

For s(n) = p^e, e > 2, (i.e., in A246549):

a(n) = A364988(s(n)) - [e mod 2 = 0]*p^(e/2)

= (p^e - 1)/(p-1) - [e mod 2 = 0]*sqrt(p^e) - 1.