A391713 Number of divisors d | n such that d is neither squarefree nor powerful (in A332785).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0

Offset

1,24

Comments

Number of terms in the intersection of A332785 and row n of A027750.

a(n) = 0 for n in A303554, where A303554 is the union of powers of primes A000961 and squarefree numbers A005117.

a(n) > 0 for n in A126706 = A000027 \ A303554, where A126706 is the intersection of A013929 (nonsquarefree) and A024619 (numbers that are not prime powers).

Links

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

Formula

a(n) = A000005(n) - A005361(n) - 2^A001221(n) + 1.

Example

Let s = A126706.
Table of n, s(n), a(n) for select n:
   n                          a(n)  A332785 intersect row n of A027750
--------------------------------------------------------------------------
  12 =   s(1) = 2^2 * 3         1   {12}
  18 =   s(2) = 2 * 3^2         1   {18}
  20 =   s(3) = 2^2 * 5         1   {20}
  24 =   s(4) = 2^3 * 3         2   {12, 24}
  28 =   s(5) = 2^2 * 7         1   {28}
  48 =  s(10) = 2^4 * 3         3   {12, 24, 48}
  96 =  s(26) = 2^5 * 3         4   {12, 24, 48, 96}
 120 =  s(35) = 2^3 * 3 * 5     6   {12, 20, 24, 40, 60, 120}
 180 =  s(57) = 2^2 * 3^2 * 5   7   {12, 18, 20, 45, 60, 90, 180}
 192 =  s(61) = 2^6 * 3         5   {12, 24, 48, 96, 192}
 240 =  s(77) = 2^4 * 3 * 5     9   {12, 20, 24, 40, 48, 60, 80, 120, 240}
1536 = s(571) = 2^9 * 3         8   {12, 24, 48, 96, 192, 384, 768, 1536}

Mathematica

f[x_] := Times @@ (# + 1) - Times @@ # - 2^Length[#] + 1 &[FactorInteger[x][[;; , -1]] ]; Array[f, 120]

Crossrefs

Cf. A000005, A001221, A005361, A013929, A027750, A052485, A126706, A303554, A332785.

Sequence in context: A357879 A072325 A294929 * A391074 A076948 A255309

Adjacent sequences: A391710 A391711 A391712 * A391714 A391715 A391716

Keyword

nonn,easy

Author

Michael De Vlieger, Mar 21 2026

Status

approved