A378899 Number of primes between successive powerful numbers k that are not prime powers (i.e., k in A286708).

11, 9, 5, 3, 6, 10, 2, 1, 1, 13, 5, 11, 1, 5, 2, 7, 3, 10, 13, 4, 0, 15, 2, 11, 4, 9, 1, 4, 13, 7, 2, 1, 9, 10, 6, 1, 2, 9, 12, 7, 4, 18, 5, 4, 17, 0, 8, 3, 13, 23, 2, 23, 10, 1, 15, 0, 7, 18, 3, 13, 7, 4, 7, 5, 4, 13, 2, 6, 10, 11, 29, 4, 2, 11, 1, 28, 2, 14

Offset: 0

Author

Michael De Vlieger, Dec 10 2024

Keywords

• nonn
• easy

Examples

			Let s = A286708.
a(0) = 11 since {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31} are primes less than s(1) = 36.
a(1) = 9 since {37, 41, 43, 47, 53, 59, 61, 67, 71} are primes that exceed s(1) but not s(2) = 72.
a(2) = 5 since {73, 79, 83, 89, 97} are primes p such that s(2) < p < s(3), where s(3) = 100.
a(3) = 3 since {101, 103, 107} are primes p such that s(3) < p < s(4), where s(4) = 108, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000040, A000720, A240590, A286708, A378699.

Programs

Mathematica

s = With[{nn = 5000},
  Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Not@*PrimePowerQ]];
{PrimePi[s[[1]]]}~Join~Differences@ Map[PrimePi, s]

Formula

a(0) = pi(36) = A000720(36) = 11.

For n > 0, a(n) = pi(A286708(n+1)) - pi(A286708(n)).