A394561 Number of divisors d | A286708(n) such that d is an Achilles number (in A052486).

0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 1, 1, 1, 3, 0, 0, 1, 2, 3, 1, 0, 1, 2, 5, 0, 1, 2, 2, 0, 1, 3, 0, 0, 4, 1, 1, 1, 0, 0, 2, 2, 5, 0, 3, 1, 5, 3, 1, 0, 3, 1, 1, 7, 0, 3, 1, 2, 1, 2, 0, 1, 2, 3, 0, 1, 0, 2, 7, 3, 3, 0, 2, 7, 1, 5, 0, 1, 0, 3, 1, 4, 1, 0, 0, 3, 7, 3

Offset

1,8

Comments

Cardinality of d in row A286708(n) of A027750 such that d is in A052486, where A286708 = A001694 \ A246547 \ {1}; i.e., powerful numbers without 1 and prime powers.

Links

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

Formula

Let s = A286708.

a(n) >= 1 for s(n) in A052486.

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 A052486
-----------------------------------------------------------
  1   0     36   {}
  2   1     72   {72}
  3   0    100   {}
  4   1    108   {108}
  5   1    144   {72}
  8   2    216   {72, 108}
 14   3    432   {72, 108, 432}
 24   5    864   {72, 108, 288, 432, 864}
 34   4   1296   {72, 108, 432, 648}
 53   7   2592   {72, 108, 288, 432, 648, 864, 2592}
104   8   6912   {72, 108, 288, 432, 864, 1152, 3456, 6912}
106   6   7200   {72, 200, 288, 800, 1800, 7200}

Mathematica

achillesQ[x_] := And[Min[#] > 1, GCD @@ # == 1] &[FactorInteger[x][[;; , -1]] ]; s = With[{nn = 2^13}, Rest@ Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3] } ], Not @* PrimePowerQ] ]; Table[DivisorSum[s[[n]], 1 &, achillesQ], {n, 100}]

Crossrefs

Cf. A001694, A027750, A052486, A286708, A376514.

Sequence in context: A053250 A364259 A302242 * A336915 A236627 A116664

Adjacent sequences: A394558 A394559 A394560 * A394562 A394563 A394564

Keyword

nonn

Author

Michael De Vlieger, Mar 30 2026

Status

approved