cp's OEIS Frontend

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.

%I A379523 #31 Jan 19 2025 09:29:50
%S A379523 6,10,12,30,54,18,30,24,30,126,30,30,42,120,238,90,60,56,42,50,126,60,
%T A379523 510,162,130,168,60,336,70,150,234,66,240,110,990,90,378,432,84,132,
%U A379523 78,112,210,270,546,90,110,456,330,150,2046,1092,182,714,102,350,260
%N 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.
%C A379523 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.
%H A379523 Michael De Vlieger, <a href="/A379523/b379523.txt">Table of n, a(n) for n = 1..10000</a>
%H A379523 Michael De Vlieger, <a href="/A379523/a379523.png">Log log scatterplot of a(n)</a> n = 1..10^5.
%F A379523 Let s(n) = A320966(n).
%F A379523 a(n) <= A364988(s(n)).
%F A379523 Using Iverson brackets:
%F A379523 For s(n) = 2^e, e > 2, (i.e., in A000079):
%F A379523 a(n) = 2^e - [e mod 2 = 0]*sqrt(2^e)-2
%F A379523      = 2^e - 2*(e-1) for even e or 2^e-2 for odd e.
%F A379523      = A364988(s(n)) - [e mod 2 = 0]*2^(e/2) for s(n) = 2^e, e > 2.
%F A379523 For s(n) = p^e, e > 2, (i.e., in A246549):
%F A379523 a(n) = A364988(s(n)) - [e mod 2 = 0]*p^(e/2)
%F A379523      = (p^e - 1)/(p-1) - [e mod 2 = 0]*sqrt(p^e) - 1.
%e A379523 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.
%e A379523 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.
%e A379523 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.
%e A379523 a(4) = 30 since s(4) = 32 = (2^5-1)/(2-1)-1 = 30.
%e A379523 a(5) = 54 since s(5) = 64 = (2^6-1)/(2-1)-(2^3)-1 = 54.
%e A379523 a(6) = 18 since s(6) = 72 = 6*12; 6 | 12 but 12 > 6; 6+12 = 18.
%e A379523 a(8) = 24 since s(8) = 108 = 6*18; 6 | 24 but 24 > 6; 6+18 = 24, etc.
%t A379523 nn = 2500;
%t A379523 s = Union@ Select[Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
%t A379523   Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] > 0 &];
%t A379523 Map[Function[n,
%t A379523   DivisorSum[n, # &,
%t A379523     And[PrimeNu[#1] == PrimeNu[#2] == #3,
%t A379523         Xor[Divisible[#2, #1], Divisible[#1, #2]]] & @@
%t A379523     {#, n/#, PrimeNu[n]} &]], s]
%Y A379523 Cf. A000079, A001221, A007947, A246549, A320966, A364988.
%K A379523 nonn
%O A379523 1,1
%A A379523 _Michael De Vlieger_, Jan 15 2025