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%I #28 Nov 30 2020 08:59:05
%S 0,6,14,12,16,10,16,66,42,10,26,70,58,14,46,86,18,114,72,74,78,72,74,
%T 96,78,14,50,76,78,130,110,286,164,170,424,154,70,132,336,162,160,90,
%U 400,342,144,36,208,108,284,98,138,216,20,66,132,504,320,120,354
%N (Smallest prime >= 5^n) - (largest prime <= 5^n).
%C Size of prime gap containing the number 5^n, for n > 1.
%C 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.
%C The arithmetic mean of a(n)/n for the terms 2..500 is 3.220 ~ 2*log(5) = A016648.
%H A.H.M. Smeets, <a href="/A338419/b338419.txt">Table of n, a(n) for n = 1..500</a>
%H A.H.M. Smeets, <a href="/A338419/a338419.png">Values of a(n)/n for n=2..500 ordered versus (n-1)/499</a>
%F a(n) = A013599(n) + A013605(n) for n > 1.
%t a[1] = 0; a[n_] := First @ Differences @ NextPrime[5^n, {-1, 1}]; Array[a, 60] (* _Amiram Eldar_, Oct 30 2020 *)
%o (PARI) a(n) = if (n==1, 0, my(pw=5^n); nextprime(pw+1) - precprime(pw-1)); \\ _Michel Marcus_, Oct 27 2020
%Y Cf. A013599, A013605, A016648.
%Y Cf. A058249 (2^n), A338155 (3^n), A338376 (6^n), A038804 (10^n).
%K nonn
%O 1,2
%A _A.H.M. Smeets_, Oct 25 2020
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