A322604 Factorial expansion of exp(gamma) = Sum_{n>=1} a(n)/n! with a(n) as large as possible.

1, 1, 1, 2, 3, 4, 2, 4, 7, 5, 6, 5, 12, 1, 12, 9, 0, 7, 4, 14, 10, 17, 2, 14, 23, 4, 2, 2, 16, 2, 10, 18, 23, 26, 26, 26, 24, 1, 17, 26, 18, 12, 0, 15, 42, 34, 39, 33, 20, 18, 40, 43, 12, 47, 51, 10, 50, 35, 14, 23, 16, 1, 55, 41, 34, 29, 14, 41, 35, 60, 53, 45, 61, 35, 49, 73, 13, 13, 57, 59

Offset: 1

Author

Tristan Cam, Dec 20 2018

Keywords

• nonn

Comments

Gamma is the Euler-Mascheroni constant (A001620).

Examples

			exp(gamma) = 1 + 1/2! + 1/3! + 2/4! + 3/5! + 4/6! + 2/7! + 4/8! + ...
		

Links

• Eric Weisstein's World of Mathematics, Harmonic Expansion
• Index entries for factorial base representation

Crossrefs

Cf. A073004 (decimal expansion), A094644 (continued fraction), A001620 (Euler-Mascheroni constant).

Programs

Maple

Digits:=200: a:=n->`if`(n=1,floor(exp(gamma)),floor(factorial(n)*exp(gamma))-n*floor(factorial(n-1)*exp(gamma))): seq(a(n),n=1..100); # Muniru A Asiru, Dec 20 2018

Mathematica

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

PARI

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

Formula

Sum_{n>=1} a(n)/n! = exp(gamma) = A073004.