A395300 Sum of numbers k < n such that rad(k) | n but k does not divide n, where rad = A007947.

0, 0, 0, 0, 0, 4, 0, 0, 0, 12, 0, 17, 0, 12, 9, 0, 0, 40, 0, 24, 9, 28, 0, 43, 0, 28, 0, 24, 0, 163, 0, 0, 36, 60, 25, 107, 0, 60, 36, 73, 0, 214, 0, 56, 52, 60, 0, 122, 0, 120, 36, 56, 0, 180, 25, 97, 36, 60, 0, 432, 0, 60, 76, 0, 25, 396, 0, 120, 36, 392, 0

Offset

1,6

Comments

Let omega(k) = A001221(k) be the number of distinct prime factors of k.

Scatterplot shows zones associated with the number of distinct prime factors of squarefree and nonsquarefree n; a(n) tends to be smaller for squarefree n compared to nonsquarefree m with omega(n) = omega(m). Likewise, a(n) tends to be smaller than a(k) for omega(n) < omega(k).

Links

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

Michael De Vlieger, Log log scatterplot of a(n), n = 1..65536 (ignoring zeros), with a color function indicating a(n) as follows: dark green = squarefree composite, light blue = neither squarefree nor powerful, and accentuating the following with larger points: red = prime, orange = perfect power of squarefree composite, purple = Achilles number.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..262144 (ignoring zeros), with a color function indicating n as follows: dark green = squarefree, light blue = neither squarefree nor powerful, and accentuating the following with larger points: bright green = primorial, orange = perfect power, purple = Achilles number.

Formula

a(n) = sum of row n of A272618.

a(n) = A244974(n) - A000203(n).

a(n) = 0 for n in A000961 (powers of primes).

a(n) = A020639(n)^2 for n in A138109.

a(n) = A020639(n)^(n*(n+3)/2) = A020639(n)^A000096(n) for n in A138511.

Example

Table of n, a(n) for select n:
   n   a(n)  Row n of A272618.
  --------------------------------------------------------------------
   6     4   {4}
  10    12   {4, 8}
  12    17   {8, 9}
  14    12   {4, 8}
  15     9   {9}
  18    40   {4, 8, 12, 16}
  24    43   {9, 16, 18}
  30   163   {4, 8, 9, 12, 16, 18, 20, 24, 25, 27}
  42   214   {4, 8, 9, 12, 16, 18, 24, 27, 28, 32, 36}
  60   432   {8, 9, 16, 18, 24, 25, 27, 32, 36, 40, 45, 48, 50, 54}
  78   476   {4, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 52, 54, 64, 72}

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Table[If[PrimePowerQ[n], 0, Total@ Select[Range[n], Divisible[n, rad[#]] &] - DivisorSigma[1, n] ], {n, 120}]
(* Alternative: first, load the regs program in Links at A369609, then: *)
Table[If[PrimePowerQ[n], 0, Total@ regs[n] - DivisorSigma[1, n]], {n, 120}]

Crossrefs

Cf. A000203, A007947, A027750, A162306, A244974, A272618.

Sequence in context: A389453 A239261 A391569 * A242707 A236379 A126849

Adjacent sequences: A395297 A395298 A395299 * A395301 A395302 A395303

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 22 2026

Status

approved