login
Number of decimal digits in the numerator of the 10^n-th harmonic number.
2

%I #24 Jun 23 2023 07:49:19

%S 1,4,41,434,4346,43451,434111,4342303,43428680

%N Number of decimal digits in the numerator of the 10^n-th harmonic number.

%C See A114468 for denominators.

%C Eric Weisstein's link conjectures that for both this sequence and A114468, a(n) ~ (log_10(e) = A002285)*10^n. - _Nathan L. Skirrow_, Jun 22 2023

%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">MathWorld: Harmonic Number</a>

%t Table[IntegerLength[Numerator[HarmonicNumber[10^n]]],{n,0,8}](* _Harvey P. Dale_, May 24 2019 *)

%o (Python)

%o from gmpy2 import digits, mpq

%o def a(n): return len(digits(sum(mpq(1, n) for n in range(1, 10**n+1)).numerator))

%o print([a(n) for n in range(6)]) # _Michael S. Branicky_, Jun 22 2023

%Y Cf. A001008, A002285, A002805, A114468.

%K nonn,base,more

%O 0,2

%A _Eric W. Weisstein_, Nov 29 2005

%E Edited by _Charles R Greathouse IV_, Aug 05 2010