A114468 -id:A114468 - cp's OEIS frontend

A002285 Decimal expansion of common logarithm of e.

4, 3, 4, 2, 9, 4, 4, 8, 1, 9, 0, 3, 2, 5, 1, 8, 2, 7, 6, 5, 1, 1, 2, 8, 9, 1, 8, 9, 1, 6, 6, 0, 5, 0, 8, 2, 2, 9, 4, 3, 9, 7, 0, 0, 5, 8, 0, 3, 6, 6, 6, 5, 6, 6, 1, 1, 4, 4, 5, 3, 7, 8, 3, 1, 6, 5, 8, 6, 4, 6, 4, 9, 2, 0, 8, 8, 7, 0, 7, 7, 4, 7, 2, 9, 2, 2, 4, 9, 4, 9, 3, 3, 8, 4, 3, 1, 7, 4, 8, 3, 1, 8, 7, 0, 6

Offset: 0

N. J. A. Sloane

Sometimes also called Briggs's constant after the English mathematician Henry Briggs (1561-1630). - Martin Renner, Jan 03 2022

			0.4342944819...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 25, equations 25:14:4 at page 232.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 79.
Jonathan Sondow and Eric W. Weisstein, MathWorld: Harmonic Number.
Horace S. Uhler, Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17, Proc. Nat. Acad. Sci. U.S.A. 26, (1940), pp. 205-212.
Wikipedia, Henry Briggs.
Index entries for transcendental numbers.

Cf. A001008, A002392, A002805, A114467, A114468.

evalf[100](1/log(10)); # Martin Renner, Jan 03 2022

RealDigits[N[1/Log[10], 100]][[1]] (* Vincenzo Librandi, Mar 25 2013 *)
RealDigits[Log10[E],10,120][[1]] (* Harvey P. Dale, Apr 17 2022 *)

1/log(10) \\ Charles R Greathouse IV, Jan 04 2016

Equals log_10(e) = 1/log(10) = 1/A002392. - Eric Desbiaux, Jun 27 2009

Conjecture by Eric Weisstein: Equals lim_{n->oo} b(n)/10^(n-1), for b=A114467 or b=A114468 (i.e., is the limit of the decimal expansion of the number of decimal digits in both the numerator and denominator of the (10^n)th harmonic number). More generally, log_k(e) seems to equal lim_{n->oo} floor(log_k(b(k^n)))/k^(n-1), for b=A001008 or b=A002805 and k >= 2. - Natalia L. Skirrow, Feb 12 2023

A114467 Number of decimal digits in the numerator of the 10^n-th harmonic number.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 29 2005

See A114468 for denominators.

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

J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number

Cf. A001008, A002285, A002805, A114468.

Table[IntegerLength[Numerator[HarmonicNumber[10^n]]],{n,0,8}](* Harvey P. Dale, May 24 2019 *)

from gmpy2 import digits, mpq
def a(n): return len(digits(sum(mpq(1, n) for n in range(1, 10**n+1)).numerator))
print([a(n) for n in range(6)]) # Michael S. Branicky, Jun 22 2023

Edited by Charles R Greathouse IV, Aug 05 2010

cp's OEIS Frontend

A002285 Decimal expansion of common logarithm of e.

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