A396831 a(n) is the n-th digit after the decimal point in the decimal expansion of the natural logarithm of n.

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

Offset

1,2

Comments

For n >= 2, log(n) is transcendental (by the Lindemann-Weierstrass theorem) and hence irrational, so its n-th decimal digit is well-defined; log(1) = 0 gives a(1) = 0.

Links

Table of n, a(n) for n=1..99.

Formula

a(n) = floor(10^n * (log(n) - floor(log(n)))) mod 10.

Example

log(1) = 0.0000........., so a(1) = 0.
log(2) = 0.6931471805..., whose 2nd decimal digit is 9, so a(2) = 9.
log(5) = 1.6094379124..., whose decimal digits after the point are 6,0,9,4,3,..., so the 5th is 3 and a(5) = 3.

Prog

(Python)
from decimal import Decimal, getcontext
def a(n):
    if n == 1: return 0
    getcontext().prec = n + 50
    return int(str(Decimal(n).ln()).split(".")[1][n-1])
print([a(n) for n in range(1, 100)])

Crossrefs

Cf. A002162 (log 2), A002391 (log 3), A016627 (log 4), A016628 (log 5), A016629 (log 6), A016630 (log 7), A016631 (log 8), A016632 (log 9), A002392 (log 10).

Sequence in context: A021897 A378205 A225458 * A092172 A133619 A276499

Adjacent sequences: A396827 A396828 A396829 * A396833 A396834 A396836

Keyword

nonn,base,easy,new

Author

Nayanesh Reddy Galiveeti, Jun 07 2026

Status

approved