A075885 a(n) = 1 + n + n*[n/2] + n*[n/2]*[n/3] + n*[n/2]*[n/3]*[n/4] +... where [x]=floor(x).

1, 2, 5, 10, 29, 46, 169, 239, 745, 1450, 4111, 5182, 27157, 33164, 84001, 186496, 610065, 713474, 3061009, 3526553, 13783421, 27380452, 63264389, 71240523, 444872761, 620729126, 1400231613, 2615011102, 9094701085, 10008828958

Offset: 0

Author

Paul D. Hanna, Oct 16 2002

Keywords

• easy
• nonn

Comments

Conjecture: limit a(n)^(1/n) = 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).

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

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

Examples

			a(5) = 1 + 5 + 5[5/2] + 5[5/2][5/3] + 5[5/2][5/3][5/4] + 5[5/2][5/3][5/4][5/5] = 1 + 5 + 5*2 + 5*2*1 + 5*2*1*1 + 5*2*1*1*1 = 46.
		

Links

• Paul D Hanna, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

Crossrefs

Cf. A207643, A207644, A207645, A208060, A085361.

Programs

PARI

{a(n)=1+sum(m=1,n,prod(k=1,m,floor(n/k)))}
for(n=0,60,print1(a(n),", "))

PARI

a(n)=my(k=1);1+sum(m=1,n,k*=n\m) \\ Charles R Greathouse IV, Feb 20 2012

Formula

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