V5115

Number of symmetric semidivisor kernels of n.

3, 17, 55, 149, 371, 862

2, 1

Symmetric semidivisor counting function.
This function ascribes to squarefree kernel ϰ = V0220(n) = A7947(n).
v(n) = 0 for prime power n ∈ V9 (n ∈ A961).

**

Table of n, a(n) for n = 1..2^14.

For ϰ = 6, we have the kernels (2,3), (3,4), (8,9). Since V0220(6m), m ∈ V5301 (n ∈ A3586) is also in V5301, the kernels for ϰ = 6 pertain as well to 6m. Therefore, for n {6, 12, 18, 24, 36, 48, 54, …} have the same 3 kernels.
For ϰ = 10, we have the kernel (4,5), which is shared by any strongly 10-regular number n.

Block[{c, d, f, K, R},
  f[n_, m_ : 0] :=
  Block[{w , lim = If[m <= 0, n, m]},
     Sort@ ToExpression@
       Function[w,
         StringJoin["Block[{n = ", ToString@ lim, "}, Flatten@ Table[",
           StringJoin@
             Riffle[Map[ToString@ #1 <> "^" <> ToString@ #2 & @@ # &,
               w], " * "], ", ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@
         MapIndexed[
           Function[p,
                StringJoin["{", ToString@ Last@ p, ", 0, Log[",
                 ToString@ First@ p, ", n/(",
               ToString@
                InputForm[
                 Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]],
                 ")]}"]]@ w[[First@ #2]] &, w]]@
        Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, #[[All, 1]]
            ] &@ FactorInteger@ n];
  {0}~Join~Table[If[PrimePowerQ[n],
    0,
    (c = 0;
    Set[{K, p}, {Times @@ #[[All, 1]], #[[1, 1]]}] &@ FactorInteger[n];
    R = f[K, (K^2)/p]; d = Divisors[K];
      Map[Function[k,
      c += Count[k + d, _?(And[CoprimeQ[k, #], MemberQ[R, #]] &)]],
       Rest@ R]; c) &, Rest@ FoldList[Times, Prime@ Range[5]]]
 ] (* Michael De Vlieger, April 20 2023 *)

Cf. V0220, V5116.

nonn,