A378875 Number of Achilles numbers k between consecutive perfect powers.

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

Offset

1,34

Comments

Within the sequence S = A001694 of powerful numbers, we have either perfect powers k (in A001597) and numbers m that are not perfect powers, i.e., Achilles numbers (in A052486). This sequence is the number of m between k.

Links

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

Example

We partition S = A001694 by numbers k in A001597 (in brackets) and derive the following irregular table:
     [1];              hence a(1) = 0,
     [4];                    a(2) = 0,
     [8];                    a(3) = 0,
     ...
    [64], 72;               a(11) = 1,
    [81];                   a(12) = 0,
   [100], 108;              a(13) = 1,
     ...
   [625], 648, 675;         a(34) = 2,
     ...
  [4489], 4500, 4563, 4608; a(85) = 3, etc.

Mathematica

nn = 2^12;
s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];
-1 + Length /@ TakeList[s,
  {1}~Join~Differences@
  Position[s, _?(GCD @@ FactorInteger[#][[All, -1]] > 1 &),
  Heads -> False][[All, 1]] ]

Crossrefs

Cf. A001597, A001694, A052486.

Sequence in context: A151843 A276422 A323069 * A325334 A280287 A147696

Adjacent sequences: A378872 A378873 A378874 * A378881 A378882 A378883

Keyword

nonn,easy,new

Author

Michael De Vlieger, Dec 09 2024

Status

approved