A075887 - OEIS

A075887

a(n) = 1 + n + n[n/2] + n[n/2][n/3] +... + n[n/2][n/3]...[n/n], where [x]=ceiling(x).

1, 2, 5, 16, 45, 171, 421, 1968, 4553, 19225, 57261, 226854, 496309, 3136420, 6764563, 24850336, 84877201, 380461599, 805949533, 4411165990, 9288196621, 48275465722, 154143694937, 527401107276, 1100708161081, 8151403215501

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) ~ L^n where L = 3.517487255902369649399793699323864170685620..., with log(L) = Sum_{k=1..inf} log(k+1)/(k*(k+1)) = 1.2577468869443696300... (cf. A131688).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = 1 + Sum_{m=1..n} Product_{k=1..m} ceiling(n/k) for n>0 and a(0)=1.

EXAMPLE

a(5) = 171 = 1 +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*3 + 5*3*2 + 5*3*2*2 + 5*3*2*2*1, here [x]=ceiling(x).

MATHEMATICA

Table[1 +Sum[Product[Ceiling[n/k], {k, 1, j}], {j, 1, n}], {n, 0, 50}] (* G. C. Greubel, Oct 11 2018 *)

PROG

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

for(n=0, 40, print1(a(n), ", "))

(Magma) [1] cat [1 + (&+[(&*[Ceiling(n/k): k in [1..j]]): j in [1..n]]): n in [1..50]]; // G. C. Greubel, Oct 11 2018