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

1,1

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.

Example

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.

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]

Crossrefs

Cf. A376281, A379336.

Sequence in context: A241041 A335169 A164936 * A102361 A280032 A227010

Adjacent sequences: A378874 A378875 A378876 * A378878 A378879 A378880

Keyword

nonn

Author

Michael De Vlieger, Jan 27 2025

Status

approved