A208060 a(n) = 1 + 2*n + 2^2*n*[n/2] + 2^3*n*[n/2]*[n/3] + 2^4*n*[n/2]*[n/3]*[n/4] + ... where [x]=floor(x).

1, 3, 13, 43, 233, 611, 4405, 10515, 64145, 218755, 1215821, 2689083, 28162105, 61179795, 307475813, 1236997051, 8042542625, 17101581699, 146671231501, 309740445795, 2415132010441, 8877053064643, 40919003272005, 85564885298027, 1068638260341937, 2783025471994851

Offset: 0

Author

Paul D. Hanna, Feb 23 2012

Keywords

• nonn

Comments

Compare the definition of a(n) to the exponential series:

exp(2*n) = 1 + 2*n + 2^2*n*(n/2) + 2^3*n*(n/2)*(n/3) + 2^4*n*(n/2)*(n/3)*(n/4) + ...

Conjecture: limit a(n)^(1/n) = 2*L where L = 2.200161058099... is the geometric mean of Luroth expansions, where log(L) = Sum_{n>=1} log(n)/(n*(n+1)) = 0.7885305659115... (cf. A085361).

Examples

			a(5) = 1 + 2*5+ 4*5[5/2] + 8*5[5/2][5/3] + 16*5[5/2][5/3][5/4] + 32*5[5/2][5/3][5/4][5/5] = 1 + 2*5 + 4*5*2 + 8*5*2*1 + 16*5*2*1*1 + 32*5*2*1*1*1 = 611.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A075885.

Programs

PARI

{a(n)=1+sum(m=1, n, prod(k=1, m, 2*floor(n/k)))}

PARI

/* More efficient: variant of a program by Charles R Greathouse IV */
{a(n)=my(k=1); 1+sum(m=1, n, k*=2*(n\m))}
for(n=0, 60, print1(a(n), ", "))

Formula

a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} 2^k*floor(n/k).