A016659 -id:A016659 - 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).

A016464 Continued fraction for log(36).

Original entry on oeis.org

3, 1, 1, 2, 2, 36, 1, 660, 1, 1, 6, 10, 1, 1, 1, 2, 1, 2, 1, 2, 2, 7, 2, 11, 1, 2, 1, 1, 1, 2, 2, 1, 76, 3, 1, 1, 6, 6, 11, 2, 4, 1, 2, 18, 1, 1, 1, 1, 1, 15, 3, 1, 24, 2, 3, 1, 13, 1, 14, 1, 4, 2, 4, 3, 3, 1, 3, 1, 3, 1, 19, 1, 2, 2, 6

Offset: 1

N. J. A. Sloane

			3.583518938456110001624954716... = 3 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + ...)))). - _Harry J. Smith_, May 21 2009

Harry J. Smith, Table of n, a(n) for n = 1..20000

Cf. A016659 Decimal expansion. - Harry J. Smith, May 21 2009

ContinuedFraction[Log[36], 100] (* Paolo Xausa, Mar 22 2024 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(36)); for (n=1, 20000, write("b016464.txt", n, " ", x[n])); } \\ Harry J. Smith, May 21 2009

cp's OEIS Frontend

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

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