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

1,1

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

Links

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

Formula

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

Example

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

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 *)

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.

Sequence in context: A127710 A137510 A247303 * A198632 A060155 A209127

Adjacent sequences: A067868 A067869 A067870 * A067872 A067873 A067874

Keyword

nonn,easy

Author

Jon Perry, Mar 07 2002

Extensions

More terms from David Wasserman, Dec 20 2002

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

Status

approved