A378699 Number of proper prime powers between powerful numbers that are not prime powers.

7, 2, 1, 0, 3, 1, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0

Offset

1,1

Comments

Within the sequence S = A001694 of powerful numbers, we have either proper prime powers k (in A246547) and numbers m that are not prime powers (in A286708). This sequence is the number of k between m.

Links

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

Example

We partition S = A001694 by numbers m in A286708 (in brackets) and derive the following irregular table:
     4,   8,     9,   16, 25, 27, 32, [36];    hence a(1) = 7,
    49,  64,   [72];                                 a(2) = 2,
    81, [100];                                       a(3) = 1,
  [108];                                             a(4) = 0,
   121,  125,  128, [144];                           a(5) = 3,
   169, [196];                                       a(6) = 1,
  [200];                                             a(7) = 0,
  [216];                                             a(8) = 0,
  [225];                                             a(9) = 0,
   243,  256, [288];                                a(10) = 2, etc.

Mathematica

nn = 2^16; s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}]; -1 + Length /@ TakeList[s, Differences@ Rest@ Position[s, _?(! PrimePowerQ[#] &) ][[All, 1]] ]

Crossrefs

Cf. A001694, A246547, A286708.

Sequence in context: A266814 A200338 A351480 * A153589 A340617 A010505

Adjacent sequences: A378694 A378695 A378697 * A378700 A378701 A378704

Keyword

nonn,easy,new

Author

Michael De Vlieger, Dec 04 2024

Status

approved