OFFSET
0,2
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).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} 2^k*floor(n/k).
EXAMPLE
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.
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 23 2012
STATUS
approved