A292621 a(n) = a(n-1) + a(floor(log(n))) with a(1) = 1, a(2) = 2.

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 139, 143, 147, 151, 155, 159, 163, 167

Offset: 1

Author

Yi Yang, Sep 20 2017

Keywords

• nonn
• easy

Comments

a(n) > c*n*log(n)*log(log(n))*log(log(log(n)))*...*log(log...(log(n))...) (k layers) for any sufficient large n, any constant c and any positive integer k.

The sum of 1/a(i) for i = 1, 2, 3, ... diverges extremely slowly.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• KeyTo9_Fans, A Chinese post discussing the sum of 1/a(i)
• Fedor Petrov, The proof of the divergence of the sum of 1/a(i), Mathoverflow, Sep 2017.

Crossrefs

Cf. A000195, A031876, A292620.

Programs

Maple

f:= proc(n) option remember;
procname(n-1)+procname(floor(log(n)))
end proc:
f(1):= 1: f(2):= 2:
map(f, [$1..100]); # Robert Israel, Sep 28 2017

Mathematica

a[n_] := a[n] = If[n <= 2, n, a[n - 1] + a[Floor@ Log@ n]]; Array[a, 62] (* Michael De Vlieger, Sep 21 2017 *)

PARI

a(n) = if (n<=2, n, a(n-1) + a(floor(log(n)))); \\ Michel Marcus, Sep 21 2017