A075885 - OEIS

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

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OFFSET

0,2

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) +...

LINKS

Paul D Hanna, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Alladi-Grinstead Constant

FORMULA

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

EXAMPLE

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.

PROG

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

for(n=0, 60, print1(a(n), ", "))