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

Author

Michael De Vlieger, Dec 04 2024

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A001694, A246547, A286708.

Programs

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]] ]