A380691 Number of divisors d | k, d < k/d, such that (d, k/d) are neither unitary nor both coreful, where k is neither squarefree nor prime power (in A126706).

1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 3, 1, 1, 3, 2, 2, 2, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 4, 1, 2, 2, 2, 2, 2, 4, 1, 1, 2, 2, 1, 2, 4, 3, 1, 4, 1, 1, 1, 3, 5, 2, 1, 2, 5, 2, 2, 3, 2, 1, 3, 1, 4, 2, 4, 2, 2, 2, 2, 1, 6, 1, 1, 1, 2, 2, 5, 2, 1, 4, 1

Offset: 1

Author

Michael De Vlieger, Feb 09 2025

Keywords

• nonn
• easy

Comments

A divisor d | k is said to be coreful if rad(d) = rad(k), where rad = A007947.

In other words, half the number of divisors d | k such that both gcd(d, k/d) > 1 and rad(d) != rad(k/d).

Divisors d and k/d have at least 1 prime factor in common and at least one prime factor q divides one but not the other divisor. Thus, the reference domain S is the intersection of nonsquarefree numbers (k in A013929) and numbers that are not prime powers (k in A024619).

Let S = { prime p : p | d } and let T = { prime p : p | k/d }. Then this sequence counts divisor pairs (d, k/d), d < k/d, such that the symmetric difference of S and T is not empty. For instance, for k = 24 = 2*12 = 4*6, where, in both cases, the product P of the symmetric difference is 3. For k = 180 = 2*90 = 3*60 = 6*30 = 10*18 = 12*15, the products of symmetric differences are 15, 10, 5, 15, and 10, respectively. In the case of 10*18, it is evident that neither rad(10) = rad(180) nor rad(18) = rad(30).

Examples

			Table of n, a(n) listing divisors d and S(n)/d for select values of n:
    n  S(n) a(n)  d*S(n)/d
  ---------------------------------------------------------------------
    1    12   1   2*6
    2    18   1   3*6
    3    20   1   2*10
    4    24   2   2*12, 4*6
    5    28   1   2*14
    6    36   2   2*18, 3*12
    7    40   2   2*20, 4*10
   10    48   3   2*24, 4*12, 6*8
   26    96   4   2*48, 4*24, 6*16, 8*12
   57   180   5   2*90, 3*60, 6*30, 10*18, 12*15
   77   240   6   2*120, 4*60, 6*40, 8*30, 10*24, 12*20
  123   360   8   2*180, 3*120, 4*90, 6*60, 10*36, 12*30, 15*24, 18*20
		

Links

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

Crossrefs

Cf. A000005, A001221, A007947, A013929, A024619, A034444, A126706, A361430.

Programs

Mathematica

Table[1/2*(DivisorSigma[0, k] - 2^PrimeNu[k] - Apply[Times, FactorInteger[k][[All, -1]] - 1]), {k, Select[Range[12, 240], Nor[PrimePowerQ[#], SquareFreeQ[#]] &] }]

Formula

Let tau = A000005, omega = A001221, rad = A007947, and S = A126706.

a(n) = card({ d | k : d < k/d, gcd(d, k/d) > 1, rad(d) != rad(k/d) }), k = S(n).

For k in S(n), a(n) = 1/2 * tau(k) - 2^omega(k) - Product_{p|k} m-1, where p^m | k but p^(m-1) does not divide k.

For k = S(n), a(n) = 1/2 * (A000005(k) - A034444(k) - A361430(k)).