A338155 -id:A338155 - cp's OEIS frontend

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

A.H.M. Smeets, Oct 26 2020

nonn

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).

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

Cf. A013600, A013607, A016659.

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

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

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

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

A338419 (Smallest prime >= 5^n) - (largest prime <= 5^n).

Original entry on oeis.org

0, 6, 14, 12, 16, 10, 16, 66, 42, 10, 26, 70, 58, 14, 46, 86, 18, 114, 72, 74, 78, 72, 74, 96, 78, 14, 50, 76, 78, 130, 110, 286, 164, 170, 424, 154, 70, 132, 336, 162, 160, 90, 400, 342, 144, 36, 208, 108, 284, 98, 138, 216, 20, 66, 132, 504, 320, 120, 354

Offset: 1

A.H.M. Smeets, Oct 25 2020

nonn

Size of prime gap containing the number 5^n, for n > 1.
From Gauss's prime counting function approximation, the expected gap size should be approximately n*log(5), however, the observed values seem to be closer to n*log(25) = n*A016648.
The arithmetic mean of a(n)/n for the terms 2..500 is 3.220 ~ 2*log(5) = A016648.

A.H.M. Smeets, Table of n, a(n) for n = 1..500
A.H.M. Smeets, Values of a(n)/n for n=2..500 ordered versus (n-1)/499

Cf. A013599, A013605, A016648.

Cf. A058249 (2^n), A338155 (3^n), A338376 (6^n), A038804 (10^n).

a[1] = 0; a[n_] := First @ Differences @ NextPrime[5^n, {-1, 1}]; Array[a, 60] (* Amiram Eldar, Oct 30 2020 *)

a(n) = if (n==1, 0, my(pw=5^n); nextprime(pw+1) - precprime(pw-1)); \\ Michel Marcus, Oct 27 2020

a(n) = A013599(n) + A013605(n) for n > 1.

cp's OEIS Frontend

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

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