A378178 Number of powerful k between consecutive perfect (or proper) prime powers.

0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 1, 4, 0, 1, 1, 0, 6, 0, 1, 3, 2, 2, 3, 7, 1, 5, 3, 4, 1, 0, 0, 2, 6, 4, 7, 2, 5, 3, 6, 1, 3, 11, 2, 2, 0, 10, 2, 13, 6, 3, 7, 2, 3, 5, 14, 7, 2, 1, 1, 3, 11, 2, 2, 17, 3, 8, 1, 11, 9, 2, 6, 0, 11, 1, 0, 20, 5, 4, 4, 24, 23

Offset

1,10

Comments

Powerful k between perfect prime powers are in A286708.

Links

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

Example

Let s = A001694 and t = A246547.
a(1..6) = 0 since s(9) = 36 is the smallest powerful number that is not a prime power.
a(7) = 1, since only s(9) = 36 is such that t(8) < 36 < t(9), i.e., 32 < 36 < 49.
a(8) = 0, since there are no powerful numbers between t(9) = 49 and t(10) = 64.
a(9) = 1, since only s(12) = 72 is such that t(10) < 72 < t(9), i.e., 64 < 72 < 81.
a(10) = 2, since t(11) < s(14..15) < t(12), i.e., 100 and 108 exceed 81 but not 121, 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@ Position[s, _?PrimePowerQ][[All, 1]] ]

Prog

(PARI) lista(nn) = my(v=select(x->(isprimepower(x) > 1), [1..nn])); vector(#v-1, k, #select(ispowerful, [v[k]+1..v[k+1]-1])); \\ Michel Marcus, Nov 25 2024

Crossrefs

Cf. A001694, A246547, A286708.

Sequence in context: A168511 A111146 A109077 * A355916 A309023 A353429

Adjacent sequences: A378175 A378176 A378177 * A378179 A378180 A378181

Keyword

nonn,easy,new

Author

Michael De Vlieger, Nov 24 2024

Status

approved