A096622 Harmonic expansion (or factorial expansion) of the Euler-Mascheroni constant.

0, 1, 0, 1, 4, 1, 4, 1, 3, 0, 2, 3, 0, 5, 14, 12, 16, 14, 7, 13, 18, 17, 19, 11, 22, 13, 13, 26, 12, 16, 2, 26, 1, 2, 28, 18, 3, 27, 31, 27, 9, 7, 37, 28, 13, 26, 2, 34, 29, 47, 49, 34, 39, 10, 0, 42, 1, 9, 42, 1, 32, 61, 23, 57, 42, 32, 2, 12, 32, 32, 48, 42, 49, 15, 14, 39, 48

Offset: 1

Author

Eric W. Weisstein, Jul 01 2004

Keywords

• nonn

Examples

			Euler gamma = 0 + 1/2! + 0/3! + 1/4! + 4/5! + 1/6! + 4/7! + 1/8! + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Harmonic Expansion
• Index entries for factorial base representation

Crossrefs

Cf. A001620 (decimal expansion), A002852 (continued fraction).

Programs

Magma

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

Mathematica

With[{b = EulerGamma}, 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 = Euler; 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

b = euler_gamma;
def A096622(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[A096622(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

Formula

Sum_{n>=1} a(n)/n! = Euler gamma = A001620. - G. C. Greubel, Nov 26 2018