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.

16, 20, 21, 48, 27, 28, 24, 25, 32, 60, 55, 39, 40, 32, 44, 45, 112, 65, 36, 84, 84, 52, 72, 35, 91, 57, 36, 96, 36, 140, 44, 63, 64, 45, 123, 40, 68, 108, 48, 85, 120, 75, 172, 96, 80, 136, 132, 56, 95, 48, 240, 49, 88, 48, 141, 92, 108, 93, 50, 196, 52, 172

Offset: 1

Author

Michael De Vlieger, Jan 18 2025

Keywords

• nonn

Comments

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.

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.

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.

Examples

			Let s = A375055.
a(1) = 16 since s(1) = 60 = 6*10; 6 + 10 = 16.
a(2) = 20 since s(2) = 84 = 6*14; 6 + 14 = 20.
a(3) = 21 since s(3) = 90 = 6*15; 6 + 15 = 21.
a(4) = 48 since s(4) = 120 = 6*20 = 10*12; 6 + 20 + 10 + 12 = 48, 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..2^16.

Crossrefs

Cf. A007947, A375055, A379752.

Programs

Mathematica

nn = 540; rad[x_] := Times @@ FactorInteger[x][[All, 1]];
s = Select[Range[nn], PrimeOmega[#] > PrimeNu[#] > 2 & ];
Table[k = s[[n]];
  DivisorSum[k, # &,
    And[1 < GCD @@ {##},
      Nor[Divisible[#2, rad[#1] ],
          Divisible[#1, rad[#2] ] ] ] & @@
    {#, k/#} &], {n, Length[s]}]