A067871 Number of primes between consecutive terms of A246547 (prime powers p^k, k >= 2).

2, 0, 2, 3, 0, 2, 4, 3, 4, 8, 0, 1, 8, 14, 1, 7, 7, 4, 25, 2, 15, 15, 17, 16, 10, 45, 2, 44, 20, 26, 18, 0, 2, 28, 52, 36, 42, 32, 45, 45, 47, 19, 30, 106, 36, 35, 4, 114, 28, 135, 89, 42, 87, 42, 34, 66, 192, 106, 56, 23, 39, 37, 165, 49, 37, 262, 58, 160, 22

Offset: 1

Author

Jon Perry, Mar 07 2002

Keywords

• nonn
• easy

Comments

Does this sequence have any terms appearing infinitely often? In particular, are {2, 5, 11, 32, 77} the only zeros? As an example, {121, 122, 123, 124, 125} is an interval containing no primes, corresponding to a(11) = 0. - Gus Wiseman, Dec 02 2024

Examples

			The first few prime powers A246547 are 4, 8, 9, 16. The first few primes are 2, 3, 5, 7, 11, 13. We have (4), 5, 7, (8), (9), 11, 13, (16) and so the sequence begins with 2, 0, 2.
The initial terms count the following sets of primes: {5,7}, {}, {11,13}, {17,19,23}, {}, {29,31}, {37,41,43,47}, ... - _Gus Wiseman_, Dec 02 2024
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 667 terms from Lei Zhou)

Crossrefs

For primes between nonsquarefree numbers we have A236575.

For composite instead of prime we have A378456.

A000015 gives the least prime power >= n.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A080101 counts prime powers between primes.

A246547 lists the non prime prime powers, differences A053707.

A246655 lists the prime powers not including 1, complement A361102.

Cf. A001597, A024619, A031218, A046933, A276781, A345531, A366833, A377051, A377057, A377282, A377286-A377288.

Programs

Mathematica

t = {}; cnt = 0; Do[If[PrimePowerQ[n], If[FactorInteger[n][[1, 2]] == 1, cnt++, AppendTo[t, cnt]; cnt = 0]], {n, 4 + 1, 30000}]; t (* T. D. Noe, May 21 2013 *)
nn = 2^20; Differences@ Map[PrimePi, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], PrimePowerQ] ] (* Michael De Vlieger, Oct 26 2023 *)

Formula

a(n) = A000720(A025475(n+3)) - A000720(A025475(n+2)). - David Wasserman, Dec 20 2002

Extensions

More terms from David Wasserman, Dec 20 2002

Definition clarified by N. J. A. Sloane, Oct 27 2023