A067882 Factorial expansion of log(2) = Sum_{n>=1} a(n)/n!.

0, 1, 1, 0, 3, 1, 0, 3, 6, 2, 5, 4, 6, 11, 4, 11, 5, 12, 3, 5, 13, 2, 22, 6, 22, 13, 20, 7, 1, 0, 1, 20, 2, 6, 4, 1, 18, 14, 35, 2, 11, 31, 16, 19, 42, 36, 41, 0, 14, 31, 25, 43, 4, 13, 34, 53, 50, 57, 2, 30, 12, 25, 45, 24, 2, 39, 57, 51, 30, 41, 65, 15, 9, 55, 23, 4, 35, 18, 77, 43

Offset: 1

Author

Benoit Cloitre, Mar 10 2002

Keywords

• easy
• nonn

Examples

			log(2) = 0 + 1/2! + 1/3! + 0/4! + 3/5! + 1/6! + 0/7! + 3/8! + 6/9! + ...
		

Links

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

Crossrefs

Cf. A002162 (decimal expansion), A016730 (continued fraction).

Cf. A322334 (log(3)), A322333 (log(5)), A068460 (log(7)), A068461 (log(11)).

Programs

Magma

SetDefaultRealField(RealField(250)); [Floor(Log(2))] cat [Floor(Factorial(n)*Log(2)) - n*Floor(Factorial((n-1))*Log(2)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

Mathematica

With[{b = Log[2]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

PARI

default(realprecision, 250); b = log(2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

Sage

def A067882(n):
    if (n==1): return floor(log(2))
    else: return expand(floor(factorial(n)*log(2)) - n*floor(factorial(n-1)*log(2)))
[A067882(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Formula

a(n) = floor(n!*log(2)) - n*floor((n-1)!*log(2)).