A322333 Factorial expansion of log(5) = Sum_{n>=1} a(n)/n!.

1, 1, 0, 2, 3, 0, 5, 4, 4, 8, 3, 3, 2, 0, 7, 8, 0, 7, 11, 1, 18, 16, 3, 10, 16, 21, 17, 13, 20, 12, 16, 8, 27, 24, 28, 12, 9, 34, 21, 3, 9, 8, 41, 42, 35, 31, 4, 4, 37, 38, 9, 20, 10, 31, 24, 34, 44, 21, 16, 19, 24, 4, 22, 22, 47, 8, 28, 26, 32, 22, 28, 56, 44, 16, 61, 38, 3, 25, 52, 35, 73, 55, 8, 42, 25, 21, 62, 61, 7, 89, 5, 74, 89, 57, 33, 60, 13, 75, 95, 66

Offset: 1

Author

G. C. Greubel, Dec 03 2018

Keywords

• nonn

Examples

			log(5) = 1 + 1/2! + 0/3! + 2/4! + 3/5! + 0/6! + 5/7! + 4/8! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for factorial base representation of noninteger constants

Crossrefs

Cf. A016628 (decimal expansion), A016733 (continued fraction).

Cf. A067882 (log(2)), A322334 (log(3)), A068460 (log(7)), A068461 (log(11)).

Programs

Magma

SetDefaultRealField(RealField(250));  [Floor(Log(5))] cat [Floor(Factorial(n)*Log(5)) - n*Floor(Factorial((n-1))*Log(5)) : n in [2..80]];

Mathematica

With[{b = Log[5]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

PARI

default(realprecision, 250); b = log(5); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

Sage

def a(n):
    if (n==1): return floor(log(5))
    else: return expand(floor(factorial(n)*log(5)) - n*floor(factorial(n-1)*log(5)))
[a(n) for n in (1..80)]