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a(n) is the nearest integer to log(log(10^n)).
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%I #26 Nov 29 2024 23:26:26

%S 1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,

%T 4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,

%U 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5

%N a(n) is the nearest integer to log(log(10^n)).

%C The sequence grows relatively slowly. For example, for n < 10^7, a(n) <= 17.

%C a(n) is roughly the expected number of Wieferich primes (cf. A001220 and Knauer, Richstein, 2005, p. 1560) as well as the expected number of Fibonacci-Wieferich primes (Wall-Sun-Sun primes) (cf. McIntosh, Roettger, 2007, p. 2091) and Wolstenholme primes (cf. A088164 and McIntosh, 1995, p. 387) with at most n digits. It is also roughly the expected number of Wilson primes with at most n digits (cf. A007540 and Costa, Gerbicz, Harvey, 2014).

%H E. Costa, R. Gerbicz and D. Harvey, <a href="https://arxiv.org/abs/1209.3436">A search for Wilson primes</a>, arXiv:1209.3436 [math.NT], 2012; Mathematics of Computation, Vol. 83, No. 290 (2014), 3071-3091, DOI:<a href="https://doi.org/10.1090/S0025-5718-2014-02800-7">10.1090/S0025-5718-2014-02800-7</a>.

%H J. Knauer and J. Richstein, <a href="https://doi.org/10.1090/S0025-5718-05-01723-0">The continuing search for Wieferich primes</a>, Mathematics of Computation, Vol. 74, No. 251 (2005), 1559-1563.

%H R. McIntosh, <a href="https://doi.org/10.4064/aa-71-4-381-389">On the converse of Wolstenholme's Theorem</a>, Acta Arithmetica 71 (1995), 381-389.

%H R. J. McIntosh and E. L. Roettger, <a href="https://doi.org/10.1090/S0025-5718-07-01955-2">A search for Fibonacci-Wieferich and Wolstenholme primes</a>, Mathematics of Computation, Vol. 76, No. 260 (2007), 2087-2094.

%F a(n) = round(log(log(10^n))) = log n + O(1).

%t Round[Log[Log[10^Range[90]]]] (* _Harvey P. Dale_, Jan 16 2024 *)

%o (PARI) a(n) = round(log(log(10^n)))

%Y Cf. A000193, A001220, A007540, A088164.

%K nonn,easy

%O 1,2

%A _Felix Fröhlich_, Sep 27 2019