A003318 a(n+1) = 1 + a( floor(n/1) ) + a( floor(n/2) ) + ... + a( floor(n/n) ). (Formerly M1052)

1, 2, 4, 7, 12, 18, 28, 39, 55, 74, 100, 127, 167, 208, 261, 322, 399, 477, 581, 686, 820, 967, 1142, 1318, 1545, 1778, 2053, 2347, 2697, 3048, 3486, 3925, 4441, 4986, 5610, 6250, 7024, 7799, 8680, 9604, 10673, 11743, 13008, 14274, 15718, 17239, 18937, 20636

Offset

1,2

Comments

Partial sums of A003238. - Emeric Deutsch, Dec 17 2014

References

M. K. Goldberg and É. M. Livshits, Minimal universal trees. (Russian) Mat. Zametki 4 1968 371-379.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. C. Read, personal communication.

Links

Joerg Arndt, Table of n, a(n) for n = 1..1000

M. K. Gol'dberg and É. M. Livshits, On minimal universal trees, Mathematical notes of the Academy of Sciences of the USSR, September 1968, Volume 4, Issue 3, pp 713-717, translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 371-379, September, 1968.

R. C. Read, Letter to N. J. A. Sloane and notes, May 1974

Formula

G.f. A(x) satisfies: A(x) = (x/(1 - x)) * (1 + Sum_{k>=1} (1 - x^k) * A(x^k)). - Ilya Gutkovskiy, Feb 25 2020

Maple

A[1]:= 1;
for n from 1 to 99 do
  A[n+1]:= 1 + add(A[floor(n/k)], k=1..n)
od:
seq(A[n], n=1..100); # Robert Israel, Aug 24 2014

Mathematica

a[1]=1; a[n_]:=1+Sum[a[Floor[(n-1)/k]], {k, n-1}]
Array[a, 50] (* Giorgos Kalogeropoulos, Mar 31 2021 *)

Prog

(PARI) N=1001;
v=vector(N, n, n==1);
for(n=1, N-1, v[n+1]=1 + sum(k=1, n, v[floor(n/k)]) );
for(n=1, N, print(n, " ", v[n])); \\ b-file
\\ Joerg Arndt, Aug 25 2014
(Python)
from functools import lru_cache
@lru_cache(maxsize=None)
def A003318(n):
    if n == 0:
        return 1
    c, j = n+1, 1
    k1 = (n-1)//j
    while k1 > 1:
        j2 = (n-1)//k1 + 1
        c += (j2-j)*A003318(k1)
        j, k1 = j2, (n-1)//j2
    return c-j # Chai Wah Wu, Mar 31 2021

Crossrefs

Cf. A003238 (first differences).

Sequence in context: A343657 A363211 A033500 * A329398 A353150 A035300

Adjacent sequences: A003315 A003316 A003317 * A003319 A003320 A003321

Keyword

nonn

Author

N. J. A. Sloane

Status

approved