A309398 a(n) is the nearest integer to log(log(10^n)).

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, 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, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Author

Felix Fröhlich, Sep 27 2019

Keywords

• nonn
• easy

Comments

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

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).

Links

• E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, arXiv:1209.3436 [math.NT], 2012; Mathematics of Computation, Vol. 83, No. 290 (2014), 3071-3091, DOI:10.1090/S0025-5718-2014-02800-7.
• J. Knauer and J. Richstein, The continuing search for Wieferich primes, Mathematics of Computation, Vol. 74, No. 251 (2005), 1559-1563.
• R. McIntosh, On the converse of Wolstenholme's Theorem, Acta Arithmetica 71 (1995), 381-389.
• R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Mathematics of Computation, Vol. 76, No. 260 (2007), 2087-2094.

Crossrefs

Cf. A000193, A001220, A007540, A088164.

Programs

Mathematica

Round[Log[Log[10^Range[90]]]] (* Harvey P. Dale, Jan 16 2024 *)

PARI

a(n) = round(log(log(10^n)))

Formula

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