A338376 (Smallest prime >= 6^n) - (largest prime <= 6^n).

2, 6, 12, 6, 30, 14, 22, 18, 32, 12, 94, 54, 52, 18, 98, 66, 84, 18, 36, 18, 30, 138, 80, 96, 30, 142, 36, 80, 52, 26, 78, 64, 126, 138, 94, 136, 162, 276, 110, 162, 206, 94, 78, 324, 186, 128, 118, 56, 102, 390, 78, 90, 18, 62, 94, 108, 220, 100, 336, 618

Offset

1,1

Comments

Size of prime gap containing the number 6^n, for n > 1.

From Gauss's prime counting function approximation, the expected gap size should be approximately n*log(6), however, the observed values seem to be closer to n*log(36) = n*A016659.

The arithmetic mean of a(n)/n for the terms 1..1000 is 3.605 ~ 2*log(6).

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..1000

Formula

a(n) = A013607(n) + A013600(n).

Mathematica

a[n_] := First @ Differences @ NextPrime[6^n, {-1, 1}]; Array[a, 60] (* Amiram Eldar, Oct 30 2020 *)

Prog

(PARI) a(n) = my(pw=6^n); nextprime(pw+1) - precprime(pw-1); \\ Michel Marcus, Oct 27 2020

Crossrefs

Cf. A013600, A013607, A016659.

Cf. A058249 (2^n), A338155 (3^n), A338419 (5^n), A038804 (10^n).

Sequence in context: A165822 A014965 A074259 * A113540 A111658 A341664

Adjacent sequences: A338373 A338374 A338375 * A338377 A338378 A338379

Keyword

nonn

Author

A.H.M. Smeets, Oct 26 2020

Status

approved