A394619 Number of divisors d | k such that d is neither squarefree nor squareful (in A332785) and k = A126706(n).

1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 3, 1, 1, 3, 1, 1, 3, 3, 2, 3, 1, 4, 1, 1, 2, 2, 3, 3, 1, 1, 6, 1, 3, 3, 2, 2, 3, 4, 1, 1, 3, 2, 1, 3, 4, 3, 1, 6, 1, 1, 1, 3, 7, 2, 1, 2, 5, 2, 3, 3, 3, 1, 3, 1, 4, 3, 4, 2, 3, 2, 3, 1, 9, 1, 1, 1, 2, 2, 7, 3, 1, 6, 1

Offset

1,4

Comments

Let s = A126706; a(n) is the cardinality of d in row s(n) of A027750 such that d is in A332785.

This sequence resembles A380691 for small terms, however, A380691(15) = 3 but a(n) = 2. See example.

Links

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

Formula

a(n) = A391719(A126706(n)) = A000005(k) - A005361(k) - 2^A001221(k) + 1 for k in A126706.

a(n) <= A376514(s(n)).

Example

Table of n, a(n) for select n:
  n a(n)   s(n)  List of d | s(n) such that d is in A332785
-----------------------------------------------------------
  1   1     12   {12}
  2   1     18   {18}
  3   1     20   {20}
  4   2     24   {12, 24}
  5   1     28   {28}
 10   3     48   {12, 24, 48}
 26   4     96   {12, 24, 48, 96}
 35   6    120   {12, 20, 24, 40, 60, 120}
 57   7    180   {12, 18, 20, 45, 60, 90, 180}
 61   5    192   {12, 24, 48, 96, 192}
 77   9    240   {12, 20, 24, 40, 48, 60, 80, 120, 240}
571   8   1536   {12, 24, 48, 96, 192, 384, 768, 1536}
.
Let b = A380691. For 2 nonzero positive numbers j and m that are not coprime, let quality Q(j,m) be rad(j) | m but rad(m) does not divide j. In A380691, we want divisor pairs (d,k/d) such that only one of Q(d,k/d) and Q(k/d,d) is true.
a(15) = 3 since, among divisors of s(15) = 60, only {12, 24, 60} are neither squarefree nor powerful.
b(15) = 2 because, among divisors pairs (d,k/d) for k = s(15) = 60, we look for (d,k/d) as described above. Only (2,30) and (3,20) qualify; note that both Q(6,10) and Q(10,6) are true and so (6,10) is not included.

Mathematica

fQ[x_] := And[! SquareFreeQ[x], Min@ FactorInteger[x][[;; , -1]] == 1]; s = Select[Range[300], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; Table[DivisorSum[s[[n]], 1 &, fQ], {n, Length[s]}] (* or *)
f[x_] := Times @@ (# + 1) - Times @@ # - 2^Length[#] + 1 &[FactorInteger[x][[;; , -1]] ]; Array[f, 120]; s = Select[Range[300], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; Map[f, s]

Crossrefs

Cf. A000005, A001221, A005361, A027750, A126706, A332785, A376514, A380691, A391719.

Sequence in context: A232353 A366987 A380691 * A304093 A238580 A035176

Adjacent sequences: A394612 A394613 A394614 * A394620 A394621 A394622

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 29 2026

Status

approved