A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

1, 2, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Paolo Xausa, Oct 25 2023

Keywords

• nonn

Comments

Conjecture: a(n) can be only 1, 2, or 3 (with the first occurrences of 3 appearing at n = 4, 9, 30, 327 and 3512).

One less than the number of prime powers between prime(n) and prime(n+1), inclusive. - Gus Wiseman, Jan 09 2025

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Paolo Xausa, 1200 X 1200 raster image of a(n), n = 1..1440000, read left to right, top to bottom, showing a(n) = 1 in blue, a(n) = 2 in white and a(n) = 3 in red.

Crossrefs

Run lengths of A362965.

Subtracting one gives A080101.

For non prime powers we have A368748.

Positions of terms > 1 are A377057.

Positions of 1 are A377286.

Positions of 2 are A377287.

For perfect powers we have A377432.

For squarefree we have A373198.

A000015 gives the least prime power >= n, difference A377282.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A024619 and A361102 list the non prime powers, differences A375708 and A375735.

A031218 gives the greatest prime power <= n, difference A276781.

A046933(n) counts the interval from A008864(n) to A006093(n+1).

A246655 lists the prime powers not including 1.

A366835 counts primes between prime powers.

Cf. A053607, A053706, A065514, A068435, A080102, A304521, A345531, A377289.

Cf. A024620, A027883, A067871, A075526, A080769, A151800, A377436, A379157.

Programs

Mathematica

With[{upto=1000},Map[Length,Most[Split[PrimePi[Select[Range[upto],PrimePowerQ]]]]]] (* Considers prime powers up to 1000 *)

Formula

a(n) = A080101(n) + 1. - Gus Wiseman, Jan 09 2025