A376281 Number of pairs (d, k/d), d | k, d < k/d, such that gcd(d, k/d) is not in {1, d, k/d}, where k is in A379336.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 3, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1

Offset: 1

Author

Michael De Vlieger, Jan 08 2025

Keywords

• nonn

Comments

Number of ways to write k = A379336(n) as a product of numbers i and j that are neither coprime nor does one number divide the other. Both i and j are necessarily composite.

Both i and j = k/i appear in row k of A133995.

Examples

			Let s(n) = A379336(n).
a(1) = 1 since s(1) = 24 = 4*6.
a(2) = 1 since s(2) = 40 = 4*10.
a(3) = 1 since s(3) = 48 = 6*8.
a(12) = 2 since s(12) = 96 = 6*16 = 8*12.
a(16) = 3 since s(16) = 120 = 4*30 = 6*20 = 10*12.
a(44) = 4 since s(44) = 240 = 6*40 = 8*30 = 10*24 = 12*20.
a(75) = 5 since s(75) = 360 = 4*90 = 10*36 = 12*30 = 15*24 = 18*20.
a(105) = 6 since s(105) = 480 = 6*80 = 8*60 = 10*48 = 12*40 = 16*30 = 20*24, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A045763, A133995, A379336, A379552, A379752, A379772.

Programs

Mathematica

nn = 500; 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 ];
Table[k = s[[n]];
  1/2*DivisorSum[k, 1 &, ! MemberQ[{1, #1, #2}, GCD[#1, #2]] & @@ {#, k/#} &],
  {n, Length[s]}]