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

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

Author

A.H.M. Smeets, Oct 25 2020

Keywords

• nonn

Comments

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.

Links

• 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

Crossrefs

Cf. A013599, A013605, A016648.

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

Programs

Mathematica

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

PARI

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

Formula

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