A378877 Let k = A379336(n). Then a(n) = sum of divisors d | k such that d neither divides nor is coprime to k/d, and k/d does not divide d.

10, 14, 14, 15, 18, 16, 22, 18, 20, 26, 21, 42, 30, 21, 22, 82, 27, 28, 24, 38, 24, 26, 25, 42, 32, 54, 33, 106, 30, 55, 50, 30, 66, 39, 54, 40, 34, 121, 32, 66, 44, 62, 45, 150, 66, 35, 65, 36, 154, 123, 42, 52, 146, 78, 35, 78, 42, 91, 46, 57, 36, 178, 36, 78

Offset: 1

Author

Michael De Vlieger, Jan 27 2025

Keywords

• nonn

Examples

			Define quality Q(k,m) regarding necessarily composite numbers k and m that neither divide nor are coprime to one another. The examples show only those divisor pairs d, k/d, such that Q(d, k/d) is true.
Let s = A379336.
a(1) = 10 since s(1) = 24 = 4*6.
a(2) = 14 since s(2) = 40 = 4*10.
a(3) = 14 since s(3) = 48 = 6*8.
a(4) = 15 since s(4) = 54 = 6*9.
a(5) = 18 since s(5) = 56 = 4*14.
a(6) = 16 since s(6) = 60 = 6*10.
a(12) = 42 since s(12) = 96 = 6*16 = 8*12, 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..21639, showing primes in red, proper prime powers in gold, squarefree composites in green (primorials > 2 with large dots), and numbers neither squarefree nor prime powers in blue or magenta, with magenta also representing powerful numbers that are not prime powers.

Crossrefs

Cf. A376281, A379336.

Programs

Mathematica

nn = 2^10; mm = Floor@ Sqrt[nn]; p = 2; q = 3;
s = Complement[
  Select[Range[nn], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &],
  Union[Reap[
    While[p <= mm, q = NextPrime[p];
      While[p*q <= mm,
        If[p != q, Sow[p*q]]; q = NextPrime[q]];
          p = NextPrime[p]] ][[-1, 1]] ]^2];
Map[Function[n, DivisorSum[n, # &, 1 < GCD[#1, #2] < Min[#1, #2] & @@ {#, n/#} &]], s]